"On the last week" or "In the last week"?

Solution 1:

According to Google Ngram viewer, "in the last week of..." is much more common. This fits with my feeling as a native speaker, too: in or during for a range of time like a week, month, or season ("in the last week of August"); on for a specific day ("on August tenth"); at for a specific time ("at 4pm").

Solution 2:

From this article on prepositions:

Prepositions of Time: at, on, and in

We use at to designate specific times. The train is due at 12:15 p.m.

We use on to designate days and dates.

My brother is coming on Monday.

We're having a party on the Fourth of July.

We use in for nonspecific times during a day, a month, a season, or a year.

She likes to jog in the morning.

It's too cold in winter to run outside.

He started the job in 1971.

He's going to quit in August.

Your sentence specifies that a plane will be arriving some time in the last week of August. Since this is a nonspecific time, you would say:

I'll arrive in the last week of August.

If, however, you do choose to say that you'll arrive exactly on the 29th of August, this is a specific day so you would say:

I'll arrive on the 29th of August.

You could, however, choose to drop the preposition (as another person answered), but only do so if you feel comfortable--people will understand the sentence both with and without "on", so you don't need to change your speaking/writing style if you want to take someone's advice.

Solution 3:

As an American, I would drop the preposition altogether. The present participle (-ing) also feels appropriate for relating arrivals and departures:

"I'll be arriving the last week of August" is what feels natural to me.

Or simpler: "I arrive the last week of August."

You could use "in" with a specific month ("I'll arrive in August.") or season, but not for a week.

Only use "on" with one specific day ("I'll arrive on August 29th.")

Solution 4:

While both would be understood, my preference between the two would be the latter, using in. I'll note that without a choice there, I would have suggested during to replace either on or in.

Interestingly, I'll contrast this with the following set:

I'll arrive on the fourth of August.

I'll arrive in the fourth of August.

In this case, on would be preferable.