In answer to your first paragraph, you may not have the option on to download the message, and to only download the headers, and also, your messages may be bigger than the default 2kb, hence the "next time you connect" sync bit. Go through the account settings again, and make sure to hit "Options", and tell it to download messages, not just headers, and maybe increase to 4kb.
In answer to the question, I'm not aware of a single mail client like that, even for the desktop. Headers are just that, headers, and contain no part of the message. You would need to have a constant connection to view email that way, and it ends up downloading the messages to the device anyway.
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