Handle keeps wont stay up, keeps turning off.

Yes you can repair this. Sounds like the valve springs and the rubber valve seats may be worn. There are repair kits available at Lowe's, Home Depot, Wal-Mart, ect. for these faucets. And they usually come with the "tool" you need to remove the faucet handle and the cap and seal nut. After that, it's pretty straightforward, and easy. Two springs, two seats, and an O ring or two, and your'e done. Just make sure to shut off the supply lines to the sink before you start and open the faucet handle to remove any pressure left after you cut the water off. Tighten the cap or seal nut securely after the new springs and seats are in, and you should be good to go.