What you are contemplating is a long trip--not just a weekend or two in the mountains. Most of those cheap tents are made for the latter, not the former.

While you may think of a tent as simply a lightweight shelter for your sleeping bag, on a long trip is it a lot more than that. You may well get caught in storm that lasts for days. I would suggest that you look at a two-man tent--one that has at least enough room for you to sit up comfortably. Because a lot of your hike will take you through more civilzed areas (you won't be carrying two weeks of food, they way we do sometimes in the mountains) weight won't be as big an issue.

Get on that is strong enough to withstand some wind, large enough that you can stand to be in it for 36 hours, and then think about weight.

That's what I would do.