Is it January's schedule or January schedule?
I also get confused with whether to include an apostrophe before "bill."
This is my January bill or January's bill.
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Both are correct and idiomatic.
Your choice depends on the construction:
January's schedule is likely to be busy.
Here January is used with a possessive apostrophe, as in winter's cold.
The January schedule this year is likely to be busy.
Here January is used as an attributive or adjectival noun to modify schedule, as in winter clothing or history lesson.
Which you prefer to use just depends on the context and your preferences.