Can I advertise the Apple Download badge on a Billboard?

See: https://developer.apple.com/app-store/marketing/guidelines/

Sure, you can. Outdoor advertising works well for promoting your product in specific geographic areas. While billboards, bus benches, and transit advertising can be very effective for the small-business owner, any successful outdoor campaign begins with your own location's signage.
Your outdoor sign is often the first thing a potential customer sees. Your sign should be sufficiently bright and conspicuous to attract attention (without being garish) and sufficiently informative to let prospective customers know what's sold there.