For the price of the hotel you could get your own nice apartment on Airbnb in the centre, worth checking out if you don't mind using it.
Don't worry about seating in the arena, all seats are good with a nice view, I usually sit on the balcony for the nice view of the ice.
Take the bus from the airport to the centre, the stop is right outside the terminal across the car park, costs a few euro and then you can probably just walk to the hotel/airbnb you booked
Try to take Baltic taxi only if you can, they are big company and salaried drivers, others might give you an unplanned tour of the city to make a few extra euros - this doesn't bother me but some it might
You can walk to Arena riga and back from the centre in like 15 mins, or take a taxi
Riga is small city so you can easily see a lot just wandering around, the zoo is fun if you take the tram to it. Old town is great to see but to eat/drink dont just stay there as its more expensive, pick an avenue walk down it and find something else!
The railway museum across the bridge from old town is interesting
Check out the 'Central Markets' some refer to it as 'Russian markets' but either way its worth alook
Eat at Lido to try latvian food
Britain 
