Eastern Oregon is basically the same weather-wise as Idaho or other western states—maybe a little wetter and not as cold. But still high desert.
My main advice is to bring a couple different outfits, even if you gotta buy cheap wool socks, polypro, fleece, and a shell at Goodwill. Once stuff gets wet, it’s tough to dry unless you have a stove.
I’ll second the advice that you truck camp rather than backpack. Gonna make life a lot simpler and cheaper. Also, there’s almost nowhere in Northside that’s far enough from a road to make backpacking worthwhile.
Just so you are aware, the Northside tag that you can draw with 0 points is a cow only hunt. Bull tags in that unit take like 3 points usually.
Hope you have a great time and find some elk!