Don't start too fast
The first floors feel easy, but your heart rate climbs faster than on flat ground. Pace yourself and push in the second half.
Use the handrail
Pulling on the handrail takes load off your legs, especially on the turns between floors.
Take two steps at a time
Most experienced runners take two steps at once. If that does not suit you, single steps at a steady rhythm work too.
Training
A few weeks is enough: stairwell intervals, lunges and squats. Most of all, enjoy it.