RDP - correct it utilizes less bandwidth via sending pixels not documents, but that's not usually the problem. If you're able to get a signal you can replicate email with documents and then process them offline, because where you get a signal is not necessarily where you want to read, write, write, i.e. work. For unreliable connections, I used online replication, offline working. I have both systems, because some tasks like web-surfing, modern websites assume broadband speeds and the RDP type connection just sends a blurry but workable screen, but for emailing I used offline replication client software, so e.g. Thunderbird, Outlook, Lotus Notes replication (all do the same basic thing).
Last week I travelled southwest USA and similarly promised work I'd check emails quasi-daily, and the signals, when I could get one, were sometimes strong, but I'd not hold a strong reliable connection for long.
Power - well I think the context of OP is not many hours of battery life needed, just some quick checking, document handling, replying. A decent capacity charged battery and carry a charged spare batter can get hours which is all someone who is on vacation and keeping ontop of email requires. Also, what is the nature of the work? I use a small netbook and a 3G USB dongle but if it was simple text email replies then a smartphone (yer over-priced Apple or your cheaper equivalents). Carry a mains charger and top-up at any civilisation stops. Solar makes more sense when someone is trying to get more time online, and as posted by others, when you have a heavy basecamp which statically points skyward and you hike lightweight.
1) before setting off get a remote-desktop capability for when you need high-bandwidth but can't see it. This can be harder than you think. Many businesses do not have this capability. There are ways of making it work but it can be long-winded.
2) before setting off, replicate all your emails on the portable device to local storage, so you can access old information and search when offline.
When travelling use either remote desktop or email replication to handle the different types of data, types of applications, the reliability and the bandwidth.
FYI in 1996 I was doing this with a Psion 3c and a Nokia PCMCIA GSM datacard!