Local equivalent to adb install
The local pm command can do this. You need the following things:


A terminal app. (adb shell will also work, but then you might just as well use adb install).
A way to get root permission on the command line (usually su)
A copy of the APK on the device, e.g. by downloading it using a web browser.


Open the terminal app, then run:

su
pm install -r -d /sdcard/download/foobar.apk


Command line options are the same as for adb install: -r will replace existing packages, -d will downgrade if necessary.