Is this cheaper than cash-rushing the 'Duct itself, or is there another reason to do it this way?
Well, if you cash-rush the aqueduct directly, then you waste the regular production of the turn. So cash-rushing something slightly cheaper is the smart move.
Got it, though I think in my current situation I may need to focus on military production for the moment, and then aqueducts, else I might not have many towns left to build in.
While this logic seems sound at first, it is less clear if you account how reaching city size effects your net output of shields and commerce. Still you would need to delay war by a few turns. Say you gain 4 shields per aqueduct and further 40 shields by cash-rushing, then it takes 15 turns to make up for the remaining 60 shields, but only a few turns to reach the same total military as you need to count in the total production, not just the increase of it.
Also keep in mind that with 16 cities(and the town of Brusa) your empire would be quite strong already. There is no immidiate need for war. I might proceed as follows:
1. Adrianople switches to knight. No universties for now.
2. All towns(except Brusa) without an aqueduct switch to aqueduct.
3. Build a few more workers, to get numbers up to 28 to 32. Ancyra might build workers before the aqueduct due to low size.
4. Cash-rush the aqueducts as is sensible.
5. Reorganize the military. Disband all warriors. That should have occured long ago. Get rid of units that are too weak to warrant unit support. This includes 3/3 spearmen and at later dates other 3/3 units, but only after your military strenght enables you to do so safely.
6. Build up your military to 8 4/4 pikes, 8 or more 4/4 MedInfs and a reasonable amount of 4/4 knights. Decide whether you want to go heavy on knights or whether you rather use trebuchets to support slow MedInfs. Decide on a numbers of units you want first and then decide in which order to build them. You want to build pikes and MedInfs first as they give you the most battle value per shield in the eyes of the AI and thus deters it. The knights are more expensive and can wait a little longer. The trebuchets will be valueable, but they will push you over unit support. So as long as you get them to the front in time, building them last can make sense. Given the unfriendly terrain trebuchets may not be a viable options for offensives, but knights will also be slown down. In the end waiting for cavalry could be the smart move. The later implies to build some knights, mainly as safekeeper of shields with low unit support per shield. Upgrade to cavalry once sensible.
7. Continue building universities etc.. Donnot expand the military beyond reason.
Waiting for AI to clears its jungle tiles may be the smart move. For a sucessfull offensive you want to have a strong military in order to not end the offensive too early. But to just deter AI from attacking you a much smaller military will suffice. Consult F3 to check if the military adviser considers your military to be superior to the Aztecs.