I feel some crucial information is missing here. MTX is used commonly for rheum conditions but swollen fingers isn’t a diagnosis. So what exactly do you have and what condition is your dr trying to treat? If it’s arthritis from an autoimmune disorder (mixed connective tissue disease)? then MTX may be appropriate. I appreciate your concern and think a conservative approach like NSAIDS makes sense to try first.
I would clarify with your dr what is causing your finger swelling and what your diagnosis is. If you can’t get a straight answer, get copies of all your lab and imaging reports and see what you find there. Also, consider getting a second opinion to clarify the issue. I have taken MTX for years and do just fine with it. It’s considered a second line drug in the arsenal against autoimmune inflammatory disorders. Let us know how this works out please and good luck.