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.