Currently, land borders of Rwanda are closed* and will remain as such for a while. So no way you can hire a car and reach Kampala from Kigali, unless you fly.
For Rwanda, vaccinated or not, Covid-19 test must be done 72h before the departure of your first flight if connecting in one ticket. Once landed you'll be tested at airport and go to quarantine for 24h in your selected hotel until the result is out (this test can be reused if you have another flight within 72h prior departing Rwanda or to go to National Parks).
Indeed, Uganda recently implemented the e-visa process replacing the VOA. Keep in mind that Uganda may impose a covid test upon arrival soon!

* Except for returning legal resident/national citizens.