mail.app
I've already disabled "IDLE Command" so it won't push emails as soon as they arrive in my IMAP (Google Apps). I have set my mail check interval 1 hour (the longest available).
I can close my Mail App after finishing with my email routine but I sometimes forget that.
Is there a plugin or a command line script to extend the 1 hour mail check interval to, say, 3 hour or 4 hours.
Short of getting a different mail client on your iPad I can get you more than half of the way there.
Set up gmail as Microsoft Exchange server (leave the domain blank, add user and password and let it try to connect. When it fails, use m.google.com for the server and you can choose mail, calendar, address data to use ActiveSync)
Once the first mail sync starts, you can then choose days to sync and which subfolders will have push enabled. The mail badge count still only reflects unread messages in the Inbox, but you will get the mail notification sound/vibration for any new push message arriving in your selected subfolders. Your iOS device will be ready to go when you tap into the folders that have new push messages.
Apple Mail's check for mail icon looks like a new mail message icon, and can be found here:
If it can't be found you can add by right-clicking on the toolbar and choosing customize:
You can then select the icon from the list and drag it to the toolbar at the top:
You can also use "Mailbox" -> "Get All New Mail" which has a shortcut ⌘ + ⇧ + N.
You don't need to synchronize. Synchronize does more than a new mail check (see this answer)
Best Answer
You could use the following command in Terminal (or add it to a shell script).
defaults write com.apple.mail PollTime 240Where 240 is the time in minutes (i.e. 4 hours).