Learn Hardcore Icelandic: Afmælið mitt er í næsta mánuði.

-ið is the definite article attached to a neuter noun. afmæli = birthday; afmælið = the birthday.
Icelandic often keeps the definite article even when a possessive is present, so afmælið mitt literally means “the birthday my,” i.e., “my birthday.”

Why is the possessive mitt after the noun? Can it go before?

With a definite noun (one that has the article attached), the possessive normally comes after: afmælið mitt.
You cannot say mitt afmælið (preposed possessive + definite suffix) — that’s ungrammatical.
If you drop the definite suffix, you can put the possessive before: mitt afmæli. This can add emphasis/contrast: “MY birthday (as opposed to someone else’s) is next month.” Both afmælið mitt and mitt afmæli are possible, but afmælið mitt is the most neutral here.

Why is it í næsta mánuði and not something with á?

For months, Icelandic idiomatically uses í: í janúar, í febrúar, í næsta mánuði.
For days of the week and many specific dates, you use á: á mánudaginn (on Monday).
For years, you typically use á: á næsta ári (next year). So “in next month” is literally í næsta mánuði.