Learn English – stairwell or staircase

No, the sentence has some errors. A corrected version of this sentence:

I am going upstairs to the 10th floor of the building.

'Upstairs' is an adverb that means "on or to an upper floor of a building."

Note that you need the definite article "the" before "10th" in order for this sentence to sound right.

I am going up the stairs to the 10th floor of the building is also acceptable, but it has a slightly different flavour: the use of the definite article "the" before "stairs" draws more attention to the act of taking the stairs, perhaps to emphasize the choice not to take the elevator.