First question - Is your truck still stock with 24v starter and glow plugs?
Alternator 1 (drivers side) charges the front battery which runs all of the truck electrical except the starter and glow plugs (stock system).
Alternator 2 (passenger side) charges the rear battery which combines with the front for 24v to start the truck and run the glow plugs.
The only electrical component required to keep the truck running once it is started is the 12v fuel shutoff on the injector pump.
So, if you run no lights, no brake lights, etc, yes you can run the truck on the front battery without the alternator for a while. Once the battery goes dead, the truck stops running.
I just rebuilt one of my alternators. The kit was only $35. The alternator is an DelcoRemy 27SI type 100.
If the new alternator is working correctly, I would move it to the driver side and then you can run the truck for quite a while. A lot of guys will do that and just manually charge the rear battery.