It is best done with the rear housing removed from the transfer case. It can then be setup in a press and an appropriate sized driver used to push the old bushing out. A driver would then be used to install the new bushing into the case.
We do not encourage replacement of the bushing without disassembly of the transfer case. The output shaft will make the old bushing nearly impossible to remove without damage to the shaft or housing. Also, when installing the new bushing, the output shaft will make it very difficult to properly install.
Why does this bushing have a hole in it but my existing one does not?
This is a service replacement part and does vary slightly from the OE. The location of the hole when installing is not critical