How can I add 120v circuits to a 240v sub-panel with no neutral?
You should contact a local licensed electrician.  You should not do what you're  attempting, as it may be a code violation and/or dangerous. 

The circuit supplying your A/C and water pump, is likely sized to only supply those loads.  Adding more loads to that circuit will likely overload it, and cause you headaches in the future.  

If you want to supply the garage, you'll want to install a properly sized feeder between the main panel and the garage. This may require an additional panel in the garage, and will require properly bonding and grounding in the garage. 

If the panel feeder is large enough to supply the garage, and all you need is a neutral.  You'll have to run a properly sized neutral, from the main panel to the subpanel. You cannot use the grounding conductor as neutral.

As it sounds like you're a bit over your head, you might want to contact a professional to do the work properly. 



As a side note... In a 120/240V single split-phase system, you should measure 120 volts between ground and either of the hot legs. You should also measure 120 volts between either hot and neutral, 240 volts between the hots, and 0 volts between ground and neutral.