Stockholm has a lot of art museums, often with interesting special exhibitions.
The National Museum (formerly free, but no longer) has art and decorative art. There are often design-related exhibitions. They begin reducing the entrance fee as the afternoon wears on, which can save some money if you only want to see part of the collection. The National Museum may well be within walking distance of where you choose to stay, but of course walking takes time.
The Ostasiatiska Museum is just south of the NM on the next island. It has both art/decorative art and archaeological displays. Just a bit farther down is the Moderna Museum.
There are several art museums on the large island, Djurgarden, which also has the Vasa Museum, the ABBA Museum and Skansen. For these you'd probably want to use transit. Although I liked them all, the art museums on Djurgarden are probably less important than the larger museums listed previously, and you will have limited time. I'm not sure there's a covered bus shelter at every bus stop, which might be sort of important in late October.
The Nordiska Museum has mostly ethnographic displays--not really my thing, but I caught an interesting exhibition of travel posters there.
Liljevalchs has contemporary art and design.
Prins Eugens Waldemarsudde has art and decorative art in a furnished home with an attached gallery. This one is a bit farther from transit, but it has both tram and bus service.
The Thielska Galleriet shows a nice private collection of Scandinavian and French art. This one is way out near the eastern tip of the island, with less-frequent public transportation (bus only). It would be important to check the schedule to be sure you don't have a long wait.
Considerably farther from the center--requiring travel by T-bana and bus and then a walk--is the Millesgarden on Lidingo. I really like it; there's usually a good special exhibition in addition to the many works by Carl Milles. However, there's considerable walking outdoors to see everything, so you'd definitely want a nice day. It's a significant time commitment to get out there, so probably not in the cards on a short visit to Stockholm.
Not as far out but equally obscurely located (requiring T-Bana plus bus but less of a walk) is the Magasin III contemporary art museum. It's open only Friday-Sunday.
Stockholm is a museum-rich city; those are just the art-related spots I'm aware of.
There are cafes/restaurants in quite a few of the museums; I assume they operate year-round but haven't verified that.
Public transit in Stockholm is quite expensive if you buy single tickets. It's definitely worth considering a pass for the days you'll be in town if you anticipate moving around the city a lot. I prefer to walk because Stockholm is so lovely, but you can't do as much indoor sightseeing if you have multiple hour-long walks each day. There's also the fall-weather factor; I've only visited in June and July.