When it comes to the question of whether he can be repaired, the answer is yes and no. If you have bought Mr. Handy then you will surely have seen that his health is going down during fights and Stimpacks cant be used to heal him like you can with regular vault dwellers. Unfortunately, there is no way to give him back any health. Once his health is gone he will die like regular vault dwellers do.
When he dies you can repair him then for 2,000 caps. It would be cool if you could just keep him repaired the whole time, but the functionality is not yet part of the game. So if you want to heal Mr. Handy then you are going to have to let him die so you can revive him.
As far as I am aware, there is no time limit to repairing him after he is dead. If you don't have enough caps at the time, you should be able to wait until a later date in order to get him repaired.