This seems like the best place to post about my Copenhagen trip.
macpete's list was very helpful in planning my trip. Himmeriget and Koelschip are must-visits. Great bottle lists and reasonable prices. At Himmeriget it was funny seeing the locals go crazy for a Monkish hazy IPA on tap while I was obsessing over the bottle list.
I would strongly recommend skipping Ølbutikken. To say the shelves were half-full would be extremely generous. Selection is almost non-existent. With no inventory and one tap line, it's hardly a "business" in any meaningful way. Kihoskh, on the other hand, was very nice: a well-curated selection with good prices. As noted by others, the bulk of the bottles are downstairs.
There's no Uber or comparable service, but biking and walking will get you just about everywhere you'd want to go.