Title:
Shooting stars is an okay title, I tried to find a deeper meaning in it and looked up what shooting stars can represent. They represent protection and good luck. If you named the story this because of that fact, I think it could be served as a pretty decent title as she is a doctor and doctors sometimes present protection and/or good luck for patients. If that's the case then this is a very fitting title, although not very eye-catching.

Description/Foreword:
Personally, the description wouldn't have brought me in because it seems too cliché. You can kind of predict that they're going to end up liking each other and it's just... not something I'd read on my own. We all know that Mina's going to change Yoongi, that's just how it is and after reading, I really think it is a bit predictable.

*Plot:
The first thing I noticed, and this is maybe because I'm studying Korean culture and the language, and am going to move to Korea for work and all, but even if you're the same age as someone or even older, it's considered rude in Korea to not use formalities if they're someone higher in rank (example would be a doctor or a government worker or if I was older than my boss, i'd still use formalities as if they were older). So, as they aren't familiar with each other, it would still be Doctor Song. Again, I'm probably just pointing this out as I'm trying to extensively learn the mannerisms, culture, and language of Korea but that was the first thing to really stand out to me.

Also, I only read up to chapter six since I don't have the time to read all of it so please excuse me if something has changed in later chapters that changes the plot, etc, and let me know if it was changed or a plot twist happens or just something like that so I know!

As for the plot, I like the hospital!au's but at the same time, I don't think there should be romance in a plot like this. I stated this below but as far as I know, doctors cannot date current patients and Mina could get fired. Maybe if they were both patients with problems, maybe then it'd be more acceptable but in this way, I don't see it. They fall in love too quickly for me and since months were skipped, I feel like two strangers fell in love and it's cliché. That's what this story kind of is; cliché.

It has it's uniqueness such as the Hamlet ideas and things talked about in the story, I wasn't expecting much of a 'detailed' synopsis of Hamlet and I loved how passionate she was about the work of literature even though Hamlet wasn't my favorite. This was definitely something unique to the story and I feel like you really knew what you were talking about when it came to Hamlet and that surprised me honestly since I did have to read it in... uh 10^th? grade.

As for why I'm saying it's cliché: the romance. Doctors are professionals and they would know they shouldn't be with patients (I stated something down below on the next section since I tried to do research on that) especially ones that are terminal so Mina should have been fired considering a co-worker (assumbly Jackson) knew about it. I do state this again but I'm a realist and if I knew one of my doctors were dating patients, I'd probably fire them as well since I feel as if that causes bias and that just isn't allowed as far as I know. But the problem with this is that romance seems to always be cliché in stories, it's almost unavoidable. I think everyone can agree that they've written a cliché romance story and if they haven't tHEY'RE LYING (well, if they're not writers maybe they aren't...).

Very good for a first story though (since you said it was your first on asianfanfics).

Writing Style/Flow:
In chapter four, instead of 'four story' building, you typed 'four storey' which just looks kind of odd. I think that was the only thing I really caught though. The grammar in general was very good and the dialogues seemed to flow nicely but I can't imagine a doctor saying a lot of the things she did to a patient.

I feel like they got close too quickly and that she fell for him too quickly and that HE fell for her too quickly as well, she's his doctor and they are friends by chapter four, by chapter five it's pretty much stated they both like each other Yoongi doesn't even put up much of a fight either especially since he doesn't seem to like majority of doctors. I even looked up if doctors are allowed to date their patients and the answers seems to be no. "The guidance, issued yesterday, tells doctors they still cannot initiate 'ual' or 'improper' relationships with current patients, but says they can date former patients, as long as they give "careful consideration" to certain factors." Yoongi is not a former patient and never will be since it's incurable.

*Characterization:
Yoongi, to me, seemed slightly out of character. I write mainly Taegi fanfics so I tend to write Yoongi the same way and, for some reason, every Suga roleplayer I meet tends to act the same way as well. When I RP Suga, people tell me that I sound like sound like some Suga they met, that I act/sound like the real Suga (that still amazes me, like uhhhh [they call me tsundere so much too, geez]), and so on. So I feel like I have Suga's personality kind of stamped into my head and in this, he just seemed off because of it. He wasn't cold to her at all minus, maybe, once where he said all doctors are the same but brushed the idea off like a chapter later. He opened up to her too fast, fell in love too fast, and it just seemed kind of out of character to me because of the stigma (this is the only word I could think of) that lies with his personality. I feel like even in real life, he'd have trouble opening up to people and would keep to himself most of the time. So to me, he seemed slightly out of character.

As for Mina, she seemed unprofessional to me as a doctor and we both know why as I've said it a few times. Now, she was definitely smart and I like that in an OC; the knowledge you gave her about Hamlet makes her stand out as a character but at the same time, I'm just not feeling her as a character. I think doctors would be sad if their patients passed away as that's a human life gone but I don't think they should be falling in love with them which is what she did. It just seems unprofessional to someone like me. The one thing that I think is VERY strong about her as a character is, well, her brains. She graduated early, dedicated herself to school in High School, and that in itself is very admirable for an OC, not many of them are like this and they SHOULD be. Most OC's are presented as ditzy, stupid 'blonde' characters or they try to be disgustingly cute. I can't even start on how many stories I've seen where the OC tries to be cute and it's just not. A lot of the stories (not all) that get featured, I'm not trying to bring them down, but they have these stupid and idiotic OC's, or even main characters who are idols, who don't know what's going on and I hate it. So I think the OC was thought out and you gave her intelligence which I will always find as an A plus.

Personal Enjoyment:
I'm not a fan of OC fanfics and the only type of fanfics I read are Taegi ones since that's really the only 'otp' I have so I don't really like reading Suga/V (too lazy to type their names, excuse me) and OC fanfics (who am I kidding, I just really don't like OC fanfics in general) but I like hospital au's and stuff like that.

Another problem with me though, if it wasn't obvious, is that I'm quite the realist when it comes to things that aren't fantasy: like dragons, kingdoms, and that kind of story or stories that can get away with bull things which is why I pointed out that doctors, as far as I know, cannot date current patients like this. 

The story, from what I read, was decent but not for me. I'm not a romance guru and it seemed a bit too cliché for my liking but I feel like I'm the odd ball out when it comes to this. Some people love clichés and romance but that person isn't really me so obviously I wouldn't enjoy it as much as someone who is in love with that genre and such.