Considering your question about ancient wells, as Russell McMahon has indicated, there does not appear to be much information on ancient construction techniques of this type. So lets conduct a thought experiment.
Currently, modern drilled caisson foundations, say, for example, 3 m diameter shafts to depths of 30 m, may often pass through water bearing strata to reach sound bearing strata, that is either rock or highly consolidated, dense soils.
Typically this is accomplished by advancing a steel casing as the drilling head extends to greater depths. Typically the shaft is filled with a dense drilling mud (sodium bentonite clay slurry) to counterbalance hydrostatic pressures (to stop water from blowing soil back in through the base of the excavation).
Returning to antiquity, consider that knowledge of the application of the wheel, lever and arch were well known at that time, and arch construction typically employed braced timber forms that required precision of design and fabrication (this is the entire basis of the load capacity and performance of an arch - its geometrical precision). It is not beyond the imagination to expect that the interlocking stone or masonry shaft could be constructed, one course at the time, as the shaft was dug by hand. Borrowing ideas from arch construction (applied horizontally rather than vertically), timber bracing/forms could be used to ensure the precision and integrity of the shaft until completed.
As the shaft neared its destination (water bearing strata), if necessary rubble or gravel (or masonry elements for that matter) could be used as ballast to counterbalance the effects of hydrostatic pressure on soils at the bottom. This ballast could be removed or left behind, as the situation required, upon completion.