I'm 20 years old, is it okay to take Finasteride for long periods of time?

Thank you for your question. You are asking if you can take finasteride for a long period of time, as you are 20-years-old, took 1mg a day for the past month, and it has worked well for you, without side effects.
I can certainly give you some guidance about this, without a submitted photo for reference, and without a physical exam. I’m a Board Certified Cosmetic Surgeon and Fellowship-trained Oculoplastic Surgeon. I’ve been in practice in Manhattan and Long Island for over 20 years. I’ve been performing hair transplants since I started my practice, and have been prescribing finasteride to male pattern hair loss patients since 1997. I’m also the founder of TrichoStem Hair Regeneration Centers, based on a system we developed for non-surgical treatment of pattern hair loss for men and women.
First of all, if you do have pattern hair loss this young, a DHT-blocker like finasteride is especially important since early onset of male pattern hair loss usually means higher DHT-sensitivity. So, it is good you are taking it, and seeing results. To continue blocking DHT formation, you need to keep taking finasteride for life. Finasteride is a long-term drug, unlike minoxidil which can have its effects wane if you keep taking it. I have patients who have been on finasteride for over 20 years.
Side effects from finasteride are actually quite rare, at about 2% of patients. It is overwhelmingly more likely that you would have no side effects, than actually suffering from them.
Finasteride is primarily to slow hair thinning, and it can have some effect on stimulating hair growth and thickening hair. To thicken thinning hair more dramatically, and stimulate growth from hair follicles that are viable but not growing, in our practice we use the Hair Regeneration treatment of PRP and extracellular matrix to thicken thinning hair and stimulate hair growth much more dramatically than finasteride alone, and it can be done with only 1-2 treatments with 3-5 years of effectiveness. For male patients your age with high DHT-sensitivity, we still prescribe finasteride as Hair Regeneration is not a DHT-blocker. This is just something to consider as finasteride slows hair thinning, but it can’t completely stop it.
To reiterate, t is very practical for you to keep using finasteride long-term to curb your progressive hair thinning and loss. Don’t worry about side effects if you don’t have them because they are exceedingly rare. I hope you found this information helpful. Thank you for your question.

As with all medications, finasteride does have side effects (low mood, sexual dysfunction etc.). It is important you are aware they exist and if you start to suffer from them then to alert your medical professional. If you have no suffered from any side effects so far then I would continue with the medication. Please schedule a review with your Doctor every 3-6 months. 

If you are having no problems there is no reason to stop.  Check in with your doctor once a year and discuss any changes or concerns you may have.