Calculate about 15 minutes by train; 4 trains per hour.
Tickets are available from windows and from vending machines.
February will be a very bad month to do spotting ...
"Flyver Grillen" to the west of the airport best is reached by car.
Approaches to Rwy 22 can be shot from underneath the approach path.
There is a free shuttle-bus running along the northern part of the airport: Its eastern terminus is a remote parking lot (staff and longterm pax). From there it is a 5-10 minute walk (direction are pretty obvious from that place).
There will bo no BIG (= "not to be missed") attractions available downtown at that time of the year. Lovely Copenhagen best is visited when the weather permits extensive use of (urban) outdoor activities.
In case the weather turns out to be veeery uncomfortable and you decide to stay airside, there is a "Quiet room" available at T2
(or is this already T1 ?). It is elevated and overlooks a bit of the apron. It is very handy to get some sleep on the couches (or "bedchairs") provided (free of charge). Buy your food at the newspaper shop downstairs and then settle into the Quiet Room afterwards. Has worked for me a number of times.
Nothing special on shorthaul (no free food etc.); good scandinavian hospitality on longhaul flights, but do not expect much chi-chi
Carpe diem ! Life is too short to waste your time ! Keep in mind, that today is the first day of the rest of your life !