Okay, folks. So here are the old month names in the Japanese lunar calendar that people in Gensōkyō most definitely use (at the very least Aya's newspapers). Not sure why modern-day moved from these proper names though. Numbers are easy to remember, sure – April is literally "Fourth Month" – but they're boring.
Month
Japanese
January
Mutsuki (睦月)
February
Kisaragi (如月)
March
Yayoi (弥生)
April
Uzuki (卯月)
May
Satsuki (皐月) or Sanaezuki (早苗月)
June
Minazuki (水無月)
July
Fumizuki or Fuzuki (文月)
August
Hazuki (葉月)
September
Nagatsuki (長月)
October
Kannazuki (神無月)
November
Shimotsuki (霜月)
December
Shiwasu (師走)