One of the easiest ways would be to convert your month column to an ordered factor and then arrange it. Or, if you only want to arrange it and not convert to a factor you could arrange it on the factor version of itself. See the answer on this SO post for a way to do it:
df %>%
arrange(factor(Reg, levels = LETTERS[c(3, 1, 2)]), desc(Res), desc(Pop))
You could use the built-in month.name and month.abb variables if your spellings match those values. Assuming your column is called month and they are the full month names something like this should work.
df %>%
arrange(factor(month, levels = month.name))