Can I convert my table lamp into a solar powered one that unplugs the panel to charge outside leaving the lamp inside?

Yes, you can. However, you need some experience with electricity first (unless you have it). Anyway, major parts of this will include a battery bank to store the power (which may end up big and bulky) and the proper panels rated for enough amps and volts to charge it. A problem with this is that the lamp you are using probably plugs it. Which means it requires 120 volts AC. Batteries provide DC. (A light bulb will work off either) It will be difficult to produce a high enough voltage unless you use some sort of power inverter. This project is sounding expensive. You might want to substitute the bulb with some white LEDs. Although it will not be as bright, you can run it off batteries and they draw much less power making the project feasible and less expensive (no inverter, smaller/less batteries) Don't forget to use proper resistance with the LEDs, they will break otherwise! Anyway, hope this helped. Best of Luck and post how well it goes!
Something like this can be bought in IKEA, they had a program like "you buy one, at they send one to an african child".