I think most museums and sights open around 9 or 10 am. Too late for you.
I would recommend to get right to the center, to check out St. Stephens Cathedral (which opens at 6 am during the week, 7 am on Sundays), stroll around a bit, it's a really nice area and have some expensive coffee.
If you have your luggage checked through to Helsinki it should be enough to be back at the airport around an hour, 45 minutes in advance.
These are the transportation options.
They run only every 30 min, and take about 20 min, so make sure to plan your return well.
Probably you'll make it to the center at 6:30, and you'll have to leave around 9:00. Indeed not much, but for me still better than hanging around at the airport.