What type of account/connection are you using? Is it actually IMAP, POP or EWS?
POP does not have PUSH, and the menu wouldn't show it.
What is the version of your Aquamail? If it is from Google Play (and not one of the -dev), - then EWS doesn't have PUSH either.
If you account is not set up as an IMAP account, but your e-mail service provider supports IMAP, you may want to set the same account as IMAP (choose "Internet Account", then choose IMAP). You may need to set it up manually.
Then you'll have the option for PUSH.
Even when you have PUSH, you still want to have "scheduled sync" enabled and set, say to 15 or 30 minutes frequency. That will prevent any type of problems of loosing PUSH connection due to network or server idiosyncrasies.