Is it possible to calculated the concentration of hydroxide ions from the known concentration of hydronium ions?
Yes, that works. In fact, some textbooks and teachers ask this type of question (asking for the lesser ion of the autodeprotonation of water) to test which students actually read the question and which gloss over details. I fell into the trap of calculating the wrong one and stopping too early, too.

So if your question had asked for the hydroxide concentration, you would indeed calculate the hydronium concentration first (because that is actually accessible) and then invert and multiply $10^{-14}$ to get the desired one as you suggest.