Depending on context "ora" can mean "hour", "time" or "now". http://dictionary.reverso.net/italian-english/ora http://context.reverso.net/traduction/italien-anglais/ora I also saw it in an expression "a che ora" which meant "at what time" but could be translated in some circumstances as "how late"
"ore come ore" seems to mean "right now"
"di ora in ora" seems to mean "by the hour"
I don't know if there is an expression that would use it to mean "pray", but I didn't find one. It is used in the expressions lunch time and hour by hour, so it is possible that it is used in an expression for the prayer times at the monastery.
They have "pregare" for the verb "to pray" in this dictionary. http://dictionnaire.reverso.net/anglais-italien/pray
No, in Italian stare is used to also mean "stay", so switching it could mean "Where do you stay now?" "Perche state con lui?" is "Why do you stay with him?"