"I eat ramen every month."
It's saying "maitsuki."
Just like with 毎年, 毎月 has two common pronunciations: まいつき and まいげつ. The latter is considered less formal than the former, so it's less commonly used than the former.
It would really make learning easier if one didn't have to dig through all the comments to get such vital information as "hey, we previously taught you to pronounce it like this, but that is another way to pronounce it, we're using here!"
It appears as though when using the kanji 毎 with another to form the "every x" phrase, the kun'yomi is used since they're technically two separate words. The kun'yomi for 月 is つき, so it's correct as it is.
"Mai tsuki" You are correct that 月 can be pronounced "getsu", as it is in月曜日. This is the on'yomi pronunciation. "Tsuki" is the kun'yomi pronunciation, which is used here.
It's まいつき (not らいつき as you wrote). This is the correct reading for 毎月
You cold not say both. By saying 毎月はor毎月が, it is like you are saying 'Every month is eating ramen'
Alternative would be ラーメンを毎月食べます。
Just as a matter of curiosity, why is RAMEN written in katakana?
Katakana is often used for adopted foreign words and ramen is from Chinese.