Arduino for Weather and Air Monitoring: Casing and Power Source
I'd say, use a IP-54 protection plastic case. The IP is a standard, that tells you, how big particles can(not) come into the case.

About powering, depends, how often can you go and switch batteries.
If you can, you can plug the cable in the case - drill the hole through and use watertight silicone plug to ensure, it is water resistant.

In case of batteries - use some LiPo batteries with DC-DC converter to minimum voltage.
Maybe you could even add a solar panel or two for addional life extension of batteries.