Hello dear friend! I am sorry for the late reply.
It is a monastic custom on Mt. Athos to welcome all who visit it. Non-Orthodox Christians are allowed to visit Mount Athos. As far as I know, the monks allow non-Orthodox full access to the grounds and rooms to sleep, food to eat, but they can’t attend the Divine Liturgy. Some monasteries do not allow at all Non-Orthodox pilgrims in the Church services and others tend to keep them in the narthex. It depends on the monastery.
Here I found an article with information for pilgrims to Mt. Athos, including the phone numbers of all the Monasteries.
Some sites that you may find useful as well are:
God bless you!