I dont think Nicholas and his family were very appreciative of the arts. He preferred the ballet and the opera. Her taste was was mainly for Galle vases and Art Nouveau items. Largely influenced by her brother. Her taste has always been questionable anyway. So no, i dont think they ever went there.
Nicholas ll is not remembered as a great collector of art. The few paintings bought during his reign are very mediocre. There are 2 paintings purchased in 1896, by Flameng, both of which are very " chocolate boxy", and which today hang in what was Alexandra's Silver Drawing Room. At least I think they still do, I may be wrong. They were bought as part of the decorations for their new apartments. Her sister Ella was responsible for decorating the apartments, so she may have chosed the Flamengs. There are a few other items in the Hermitage today bought by Nicholas, but nothing to blow one away. A few Renaissance painting as well, I seem to remember.
Other than that, the Faberge Eggs stand out as important " objects" related to his reign.
I would not imagine that the family spent much time appreciating the beautiful paintings and objects in the Hermitage. Judging by the records I dont think that they would have walked around appreciating the Royal Collections. They were hardly ever there anyway. I cannot remember ever seeing any photographs of them in the Hermitage Museum part of the building, or anyone expressing any appreciation for items in the collection. Certainly not in the letters or diaries.
On the rare visits to the Winter Palace theatre, I would imagine that they went by carriage, as it is the building furthest away from the private apartments, which were at the opposite end of the building. To walk there through the building to get there would not be possible, its quite far.