Hook it up and see if it works. "Deep cycle" means it's kinda built to tolerate being discharged to nearly dead. No lead/acid battery really benefits from that kind of behavior. They all create sulfates that clog the battery plates as they discharge. But, if you're going to power a trolling motor, "deep cycle" is what you want 'cause you're bound to fish until the motor won't run any more.
Automatic chargers sometimes won't even kick in to charge if the battery is lower than it's built to detect and manage. In other words, you know the battery is extremely low, you know your charger is good, yet if it won't start the charging process, then it's just not built to deliver a charge when the battery voltage is really low.
If that happens and if you want to be able to use your 1 amp charger, then first hook it to a "manual" type charger. Or, hook it to your car using jumper cables. Run the engine for about 15 minutes. Then hook your 1 amp automatic to it and see if the little automatic will then take over.
People have the same problem with all batteries if the charge gets really low. Some automatics won't start a charge cycle in that case. There are some Shumacher's that will do it. They are more toward the top of the line of the truly automatics. There are also battery chargers that have both manual and automatic settings.
1. Hook it up and see what happens. Can't hurt. If it starts the charge cycle, let it run. However, 1 amp isn't much. It might not even be enough to overcome the internal resistance of your battery in such a state of discharge.
2. Hook your battery to your car with jumper cables (after starting your engine.) Let the engine run 15 minutes. Then, see if your 1 amp charger will do take over from there.