If the a/c unit requires 20A at 240V, you will need to pull a new 12/2 cable to this location, put in a 240V two pole breaker, and install a NEMA 6-20 receptacle. This will probably mean a bit of rework for the existing circuit as well, since it won't be used at this location any longer, but may need to be rerouted to work with other receptacles.

Without more info about your husband's skill level, I can't answer whether he's up to installing a new circuit or not.

Replacing the outlet and breaker is easy. The breaker literally pops into place and you just slide the wires into the slot and tighten the screw. (Note: turn the power off at the main disconnect before working in the panel.) You need to make sure you have room for the new breaker though because it will be twice as wide as the current one.

Unfortunately, that is only half the job. Presumably, they ran 14 gauge wire to match the 15amp breaker. If so, you'll need to run a new 12 gauge wire from the breaker box to the outlet. That means fishing wire up through the walls and through whatever obstacles are between that and the panel. Depending on the layout, that could be relatively easy or a total pain.

On the plus side, this is about the simplest circuit you could start out with. Just 1 wire from an outlet to the panel. The only real challenge is getting that wire from point A to point B.

The 220v will require a double breaker, most likely, or two slots in your panel. Hopefully you have room already for this.