Is there an iPhone app that can give me a push notification of a new email?
I think you might be confusing "push" and "SMS".

Push allows an email client to receive a message shortly after it has been sent, without waiting for a "fetch" window at n minutes.

If you just want all mail to show up as SMS (text message), you can use Gmail's built-in forwarding to achieve this.  It will forward the email as soon as it is received, and all SMS messages are delivered in a "push"-y manner, so this should do what you want.


Open Gmail's "Mail settings"
Select the "Forwarding and POP/IMAP" tab
Click the button to "Add a fowarding address" and enter your phone's SMS text number (you can find a very comprehensive list of carrier email-to-SMS addresses here: http://www.emailtextmessages.com/)
Verify that you own the address (they will send a verification code to the address, and you'll enter it in Gmail)
Select to "Forward a copy of incoming mail to" that address you just entered
Choose to "keep Gmail's copy in the inbox" if you would also like to be able to access your email the traditional way


You don't need to sign up for any third-party service, it's free, and you'll get your SMS messages immediately.

If you only want to forward a subset of your email, you can set that up in Gmail as a filter with a rule that forwards to your SMS email address.

Keep in mind the limitations of SMS, though.  The theoretical maximum length for an SMS is 255 concatenated 160 character messages, but most carriers will limit it to only a few messages -- so you might lose a lot of body text on longer emails.  But it will serve as good, fast notification that a new email has arrived.