Hi Christine,
l have exactly the same issue with some galvanised hive roof tops that l salvaged from some rotten hives last year. I’ve now made new roofs with built in insulation and will fix the ‘tin lids’ back on - painting them first.
From experience of using various types of galvanising paint on all kinds of things, l find that the most ‘zinc rich’ paint is one traded by Teamac. It’s called a primer but will perform equally well as a long lasting topcoat with similar properties of the original zinc galvanised metal sheet - and look good too. Some light rubbing down of the rust spots to smooth them, a good soapy wash and rinse, then these areas neutralised with oxalic type ‘rust killer’, is all that’s needed before application. Although some Teamac thinners is technically required for cleaning up, you could use a disposable brush or lots of dish soap/cream etc. Depending on the degree of rust you could just touch them up. I will be painting the whole roof inside and out for appearance and permanence prior to nailing them back on. Best not to apply in sun or wind as the solvent in the paint evaporates more quickly and the surface will drag - apply fairly quickly, try not to go back.