I have never been a monk, but I did stay in a Coptic monastery for about two weeks a few years ago in order to do linguistic research there. To the best of my recollection, for any overnight stay, a letter of permission is to be sent with you from your parish priest, or if possible your bishop (as you need to be vouched for in any case when in another bishop's territory) before visiting the monastery. Seeing as you are not Oriental Orthodox, I don't see how you could get such a letter to begin with, although I also don't know if there might be some agreements in place between Eastern Orthodox and Oriental Orthodox at this level, as I know of monks who have been received from EO monasteries into the Coptic Orthodox Church (the famous modern anchorite Fr. Lazarus El Antony is one such case; before becoming Coptic Orthodox he had been with the Serbian Orthodox). This would be something you should check with the relevant parties.
In a more general sense, it is expected that you should have some kind of connection to the particular monastery in order to be considered to take vows there. In the same way that you would not be able to show up to any EO church without proper introduction and expect to take communion, it is not to be expected that anyone who simply shows up to the monastery and expresses a desire to be a monk will actually be accepted.
I have heard of other guidelines with regard to the life you will be surrendering (that a candidate not be in financial debt, for instance), but I don't know the specifics of how things work out in any individual situation. I imagine there is a lot of variation. From my brief discussions a few years ago in the Coptic monastery with a monk there, it is far more often the case that people will come intending to live the life of a monk and eventually leave after finding it too difficult to adjust to in different aspects. Even from my ~2 weeks there, I can understand. It is a very, very different life, and requires a certain temperament and inclination that I am sad to say I do not currently possess, although I think I got quite a lot out of it as a pilgrim of sorts.
Perhaps visiting an EO monastery as a pilgrim for a little while will help you to get a small taste of the life, and then if you find yourself wanting to move forward you will be able to ask the brothers there about the specifics of how you might join the monastery on a more permanent basis.