Culture Day is a Japanese national holiday held on November 3.
Are there some special events related to this day in Osaka or Kyoto?
Is it also suggested to avoid main attractions like the Osaka Aquarium Kaiyukan or the Universal Studios Japan?
There are no special "celebrations" of the Culture Day for regular people to enjoy. But as in case of most regular holidays in Japan, some events are scheduled for this day.
Some events that deal with "culture" :
in Jōnangu Shrine in Kyoto a staging of traditional poetry reading and leisure in Heian-period attires called Kyokusui-no Utage is staged.
in Kasuga Shrine in Nara a "gagaku" traditional dance and music event is held.
in Hikone (1 hour ride from Kyoto on a special rapid train) a samurai warrior parade takes place.
Of course you should expect an increased number of people at the regular destinations you mentioned, especially if Nov 3rd falls on Monday or Friday making a long weekend.