If I rent a movie for $3.99, how long can I keep it, like if I want to watch it now and then later tonight can I do that?

For information on Apple TV rentals, see the following from: Rent movies from the Apple TV app - Apple Support

You have 30 days to start watching your rental before it's deleted from your Library. After you start watching, your rental is available for 48 hours. Learn how to change the subtitles or audio language for your rental.

For 30 days from when you rented it, or 48 hours(24 in the USA) after you started playing it, whichever comes first.

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You have 48 hours to watch it any time after you hit the play button for the first time on that movie.