You could certainly get a visa and go landside, but bear in mind that sunrise is about 6.15am in Dec/Jan, so it won't be bright and sunny until around 7am.

Go outside, take some pics of buses and taxis ?

Also, although DOH is primarily a transit hub, there is still a lot of intra-Gulf business traffic and you may find some queues when you go back in. Good news is that gates only close 20 mins before departure, so you have some time.

I've transited DOH many times but have only visited once, so I can't give much definitive advice from personal experience.

>> Especially for someone arriving from Sweden in late December
It will be less than 20 degrees C ... brrr