Hi phoinex,
Out of curiosity, do you have the grips for the pistol?
It looks to me like this should be the H&R Young America which were manufactured from 1884 - 1941. Since 1.5 million were made only the book will tell us when this one was made and this IS crucial since the early ones were made for black powder and could definitely NOT safely fire a today's .22 cartridge.
At the 311,000+ serial number You 'should' be into the smokeless powder era. We'll need confirmation from one of the guys here who have the books.
Looks to me like this is chambered for .22 short. You could go to a gun shop or ask a friend who has a .22 long cartridge and see if when inserted it extrudes past the cylinder. The pistol will MOST DEFINITELY NOT fire .22 long rifle cartridges. Finding .22 shorts may not be easy.
First, you restore it, clean it well. I suggest you do not take out the spring in the grip, it's a bit of a pain to get back in and you don't want to add extra stresses to it. These are old parts. When you do have it put together you have to dry fire it with no rounds. You need to check that the cylinder lines up with the bore for the complete revolution of the cylinder. This is 'checking the timing'.
These are all certainly wonderful little pistols and have an interesting history behind them. I have my little .32 S&W hammerless US Revolver Co. pistol in my collection and it shoots fine.
These are double action only pistols. You won't be able to cock the hammer back then squeeze the trigger. At anything over 5 yards the accuracy will generally suck raw eggs.
If you have more questions, let me know. I will send some personal messages to guys here I communicate with and help them get you the info you need. I can also help you with where to find parts.
If you don't have the grips and there is something else wrong that needs a part, the value of the pistol is very little. I paid 38$ for my .32 top break last year with nice patina to the metal and fully functional.
With no grips, and not working you can put it up for auction as 'parts not working' and you might get 50$ for it.