There's a really nice swedish speaking cat forum at Svenska katters diskussionsforum • Index
where he can probably find people locally who'd be willing to cat-sit or give advice how to find one. (Like this forum, they can be a little... obsessive
- it can be offputting to complete cat newbies who ask apparently innocent questions, like "should I let my cat outside" or, well, you know what I mean).
I got my cats privately but via the big Stockholm cat shelter, Stockholms Katthem
- they have not only the cats in the shelter currently, but also private placements listed there, which is how I found mine, then we made arrangements directly. There's always plenty of private placements there (on this page: Omplaceringar - Stockholms Katthem
- it's a bit hard to find on it's own) The girl at the top would be pretty ideal for instance: 8 years old, wormed, neutered, chipped, vaccinated, and insured, and they describe her as calm and a bit of a couch potato.
Outdoor cats in an apartment building are pretty unworkable, but taking your cat for a walk in the building yard on a leash is quite normal in Stockholm (my cats came leash trained, or as near as any cat ever gets, and they were central city apartment cats). He should check about netting on the balcony - it is required if you have a cat allowed on the balcony in some places, depends on the area - it's pretty easy to put up. The forum I put at the top will probably be able to help with that too.