Can i use fingertip press-ups to train my fingers for climbing?

You can, but you would be better doing what the traditional karateka do in Okinawa... Carry about big, water filled stone jars, (nigri game.) Press ups will exercise the muscles involved in extending your fingers more than the ones involved in gripping.

http://www.ykkfindia.org/supptraining.htm copy

The muscles for your fingers are for the most part actually in your wrist, at least with the finger flexors. Pull heavy things (like bars of various thicknesses), squeeze grippers, and try a gyroball. The climbing suggestion is good, although it is harder to measure a lot of the time.

Someone with weak hands needs to be careful not to overdo it. If his ligaments are weak, progressively harder training could stimulate them to become better, if he's eating well.
http://trickstutorials.com/forum/showthread.php?p=861522 copy