Strat-O-Matic Baseball 365

Is there such as needing to rest a #1 catcher for a day after x number of games in a row? Will the "engine" rest my #1 automatically? Is catcher fatigue a thing?

Rest is not a thing for any player or pitcher. Play them until they get injured. Then they can rest.

As LMBombers stated above, there's no reason to rest a position player; injuries will take care of that, and catchers will typically get injured more than the other positions.
However, pitchers do in fact have to rest. A starter with a * can start every 4th game, others only every 5th game. And a reliever who pitches 2 games in a row will be fatigued for the next game... and the "engine" will try not to bring him in when he's fatigued.

Pitchers do not need extra rest beyond what the game enforces. Your starters don't get better if you rest them longer than the minimums of the game rules for instance. Pitch them as often as you can.