This should be a relatively simple issue to isolate. You start with the blown fuse. It means there is something on that circuit that is shorting out. That fuse protects the following:
Alternator, Charging System Light, Cruise Control Main Switch, Cruise Control Unit, Electrical Load Detector (ELD) Unit, Evaporative Emission (EVAP) Canister Vent Shut Valve, EVAP Bypass Solenoid Valve, EVAP Purge Control Solenoid Valve, Engine Mount Control Solenoid Valve, Powertrain Control Module (PCM), Primary and Secondary Heated Oxygen Sensors, Vehicle Speed Sensor (VSS) (with Navigation System), Navigation System (2000-2004), PGM-FI
If I had to make a purely hypothetical guess, I would guess something in the alternator is shorting out but it could be any of the above components. The reason I suspect the alternator is because I have had this happen to me on 2 different Honda J35 motors before... the alternator shorts out and then even if you try to charge the battery it doesn't work when in theory you should be able to run with battery alone until it is drained.
The easy way to check this is to just fully unplug the alternator (there are two connections... one electrical connector and one bolted on + terminal). Then charge up your battery completely and start it and see if it stays running longer than a minute.