毎月 まいげつ means every month. So in your original sentence it said tomorrow is every month's exam which while effective in communicating, it does not sound natural.
Also 毎月 is a Chinese compound consisting of the modifier 毎 and noun 月. But it is imported whole as a whole word in Japanese. Usually these things have very restricted usage in Japanese as there are always Japanese native alternatives for these basic ideas which is preferred as in this case. In Chinese 毎月 would have no such restrictions.
月に means monthly. Think of に as the ly in monthly (for the purpose of understanding this example). You can also say 週に to mean weekly. 一日に for daily
However unlike English you have to follow these expressions with how many times in a month or week or day as by themselves they don't imply once or twice or however many unlike in English so that's what the 一度 is for.
月に一回 once a month or monthly
月に三回 3 times a month
週に六回 6 times a week
Etc
There are alternatives.
月毎に つきごとに
一ヶ月に付き いっかげつにつき etc
Welcome to the world of Japanese where there are a million ways to say the same thing.