I don't think any app can do what you want.
The problem is that you want to receive an SMS when a mail is received by the server rather than have the phone polling the server or push turned on. So by definition only the server knows the mail has arrived, not the phone, and so this can't be fixed by an app on the phone.
Hence I'm afraid that one way or another it's a matter of finding a service that will send a text when a mail is received (and if necessary forwarding mail received by other accounts to it) or biting the bullet and turning on push or poll.