Wire brush then Rustoleum red oxide primer 2 coats and they should be fine.

Just like Millworkman said...good prep, primer, and paint.

I much prefer the conversion coatings that dissolve the rust and them form a primer base for paint (using the iron from the now dissolved rust).

They work very well on lightly rusted things like lintels.

If there is so much rust it is flaking off, you need to remove the loose rust and then use the coating.

I've used this product:

Here is a link that might be useful: Rust Converter

Are there specific conversion products that are recommended? Also, is it advised to first strip the existing pain from the lintels as well?

Lastly, it appears there is some rust on the top part of the lintel - that contacts the brick. I obviously can't get in there to paint that. Should I try to spray anything into the gap? Or just leave it be? Thanks.

Anyone have any thoughts on whether there is anything I can do to prime the top part of the lintel that contacts the brick?

I would not concern yourself with it as it will not be easy and taking it apart to prime may prove to be a disaster.

+1

Don't even worry about it. Even if you could, the brick would very likely knock the paint off.

Just clean, prime and paint.