TL;DR: Don't worry about it.
I've stayed in lots of temples and shrines in Japan, and their accommodations are effectively identical to "secular" Japanese inns: for example, you'll be offered alcohol with meals, and boisterous drinking parties for traveling groups are not uncommon, particularly in Shinto shrine lodges. Of course, the usual rules of Japanese indoor etiquette apply, so take off your shoes before walking on tatami etc.
Only a few points may remind you that you are, in fact, in a temple:
- The staff are mostly (but not necessarily entirely) monks.
- The food will mostly likely be vegetarian.
- You may be offered the chance to join the morning service (Shingon sect) or meditation session (Zen sects like Soto/Rinzai). If you accept, and you should even though it usually involves waking up at 5 AM, literally all you need to do is sit there quietly and watch.