I would say yes. I have a 22k window unit in a 20x25 metal shop building, and another 22k window unit in a ~900sq ft mobile home. Either one is fine in all but the hottest conditions. But, here in San Antonio, that's most of the summer. Daytime highs are 90+ for 3 months, and there have been years when it crested 100 for 3 months straight (didn't happen this year). If you live somewhere easier than this i would say it will be 'ok'. I wouldn't expect the upstairs space in the cabin to be comfortable, though. It takes an absurd amount of downstairs cooling to cool an upper floor, like get the downstairs to 65 so the upstairs can stay below 80. If you want to sleep up there I would install the window unit on the upper floor and it will technically do a bit better all around. Cool air will cool the upstairs then 'fall' downstairs, and the unit will be sucking in the hotter air of the building vs the cooler air. Having the air inlet AND outlet be no more than waist high is a nasty compromise in most window unit installs.
The mobile home is insulated better than the shop, but i wouldn't say 'well'. If i leave the 22k cranked all day i can get the large open area it's in down to 69 even when it's 100 outside. There's not enough air circulation in a long skinny rectangle building to have it cool the whole thing evenly unless i run a bunch of fans to move air around. I have 2 smaller window units (one at each end) to alleviate this, but if it were distributed through good ducting like a central air unit the 22k by itself would be nearly sufficient.
The 20x25 shop is insulated, so not 'just' a metal panel building, but still has an uninsulated 12x12ft roll up door, a couple of windows, etc. Compared to most habitable structures still poorly insulated. That thing I can't get the interior below 80 when it's 100 outside unless i run the unit all day. If i let the interior get up to ~95 and then turn on the ac it won't get it below 80 before the 'sun load' starts to drop off anyway. If i turn it on in the morning and the interior never gets a chance to heat up, it will keep the interior below 80.