The fish will most likely die in such a small environment. Buy him a 10 gallon if you intend to keep for longer than two months.
Goldfish are very messy fish, and they easily grow over 10 inches, so you can quite easily figure out that a 1.5 gallon isn't by far enough for a goldfish. The absolute minimum size of a tank for a goldfish is 10 gallons and 20 for bigger varieties (then you add another 10 gallons per fish). Anything smaller, and the fish will gradually poison itself in the ammonia it excretes.
A filter for a goldfish tank should be rated for a tank at least twice the size, so if you have a 10 gal tank, think that you need a filter worthy of a 20 gallon. Smaller filters would get clogged with dirt very fast, and it wouldn't be able to deal with all the waste the fish creates.
But if I was you, I'd buy myself a test kit for pH, ammonia, nitrite and nitrate. Test the water daily, and do small water changes whenever the ammonia levels get too high. It should say on the test what's acceptable
Change no more than 10% of the water at a time! While you'd do this, be on the look out for a bigger tank for the fish
Nothing smaller than a 10 gallon! Which are usually very cheap anyway d: