Learn Hardcore Japanese: raigetu yatin wo hayame ni haraimasu.

早めに is the adverbial form of 早めだ (“somewhat early”). It means “a bit earlier than usual or earlier than something is due.” In 来月家賃を早めに払います, you’re saying “I’ll pay next month’s rent somewhat in advance (before the due date).”

How is 早めに different from 早く?

早く (hayaku) comes from 早い (“early/fast”) and usually means “quickly” or “earlier.”
早めに adds the nuance of “with extra margin” or “slightly before the normal time.”
So if you use 早く, it sounds more like “I’ll hurry up and pay,” whereas 早めに emphasizes “I’ll pay ahead of schedule.”

Why is there no particle after 来月?

Temporal nouns like 来月 (“next month”) can function as adverbials without a particle. It’s common to drop に when you’re only specifying a general time. You could say 来月に家賃を払います, but omitting に is more natural in everyday Japanese.

Can I reorder the sentence as 来月に家賃を早めに払います or 来月早めに家賃を払います?

Yes. Japanese word order is quite flexible for adverbials:

来月に家賃を早めに払います (adding に)
来月早めに家賃を払います (dropping に)
家賃を来月早めに払います
All are grammatically correct. You move elements around to change emphasis or style, but the core meaning stays the same.

Could I use 前もって instead of 早めに here?

Yes. 前もって also means “in advance,” often in more formal or written contexts. For example:
来月家賃を前もって払います
But 早めに is slightly softer and more conversational.

Why is 家賃 followed by を and not に?

The verb 払う (“to pay”) takes a direct object marked by を: you pay something (家賃を払う).
If you wanted to say “pay to” someone or someplace, you would use に (e.g. 大家さんに家賃を払う, “pay rent to the landlord”).

What’s the difference between 払います and 支払います?

Both verbs mean “to pay.”

払う/払います is the everyday, conversational form.
支払う/支払います is slightly more formal or businesslike.
In this sentence, 払います is perfectly natural for daily conversation.