I have a rusty water pipe underneath the kitchen sink that has a pinhole leak, is there anything that I can do to fix it until I can get a plumber?

If it is a small leak you can buy a repair clamp to but over the leak and tighten it up. That should last for quite sometime, I mean maybe years.
Make sure you get the right size for the pipe you have and make sure you smooth the pipe so that the clamp can seat properly

Letmetellu gave you great advice about a saddle clamp.(See image) Let me add another option to his answer. They make a repair tape for problems such as yours. To check it out click on; http://www.plumbingstore.com/repairkit.html.
Good luck, Tom