Learn Spanish (Latin America): Estuve allí un mes para asimilar la gramática. - I was there for a month to assimilate the grammar.

Yes, but it changes the meaning. Tardé un mes en asimilar la gramática means "It took me a month to assimilate the grammar," focusing on the time required to complete the task. The sentence above, Estuve allí un mes para asimilar..., means "I was there for a month in order to assimilate...", focusing on your physical stay and your purpose for being there.

Why isn't there a word for "for" before un mes? Shouldn't it be por un mes?

You can use por or durante here, but it's completely optional and often left out in everyday Spanish. Just placing the time duration (un mes) directly after the action is the most common and natural way to say "for a month."

Why is the verb estar in the preterite tense (estuve) instead of the imperfect (estaba), since a month is a continuous period of time?

Whenever an action has a clearly defined beginning and end limit—like un mes—it is viewed as a completed event in the past. Therefore, you must use the preterite estuve. You would use estaba if you were describing an ongoing background state without focusing on its timeframe (like "I was there when the phone rang").

Why do we use para instead of por before asimilar?