While Samsung stopped churning out Android updates for the original Galaxy Tab P1000 tablet, XDA has been actively releasing ROMs for it. The latest one brings Android 5.1 Lollipop update to it.
Here are the installation instructions on how to download and flash Android 5.1 lollipop on the P1000.
First download the ROM from here and put it onto the external SD card of P1000. Then go into custom recovery of P1000 and flash the ROM from there. The ROM has no gapp but can be easily flashed from here.
This ROM is made by Omni team which is behind many great firmwares for different devices including Galaxy Tab P1000.