How long will a battery last based on these numbers
There was a lot of babble in your question so it's not really clear, but you seem to be asking how long a 3.5 V 1 Ah batter will run a 24 V motor.

The answer, is no time at all in a practical application.  The motor might be able to spin with 3.5 V applied and no load, but the torque will be so low as to be useless.

There are many types of motors and types of batteries out there.  Find a combination that is properly matched.  For example, you should be able to find 12 V motors that will run well from ordinary car batteries.

However, even this is premature.  You first have to find out what the parameters are of whatever mechanical thing you are trying to drive with the motor.  In particular, find how much power it takes.  Power can be converted from one torque x speed combination to another with gears, but they can't change the overall power.  You have to start with a motor that is sized for the load.