I have personally not stayed in a monastery anywhere but I have met people who got to stay in a monastery basis there relation with the monks and if they were ready do to menial tasks in the monastery... but building this relation with monks might take time and a lot of trust the easier way ofcos would be to just visit the monastery spend the day and stay in some hostel/homestay near by... If you want a local experience homestays will be a better option but in hostels you could get to meet some really interesting personalities