The clock can certainly be run by only winding the time train. That's the one on the right. It would be a good idea to clean the movement and lubricate the pivots and escapement. Do you know anyone who works on mechanical clocks? A clock repair shop maybe?
edit: I corrected a confusing description of which side to wind. Also if it runs now you could just use it as is. Or I could walk you through what should and should nor be lubricated, amount and kind of oil.