According to police.hu info updated 22/July
You can enter Hungary by road, rail and waterway from Croatia, Austria, Romania, Serbia, Slovakia and Slovenia without any epidemiological restrictions, regardless of your nationality and coronavirus immunity, but you must have other general conditions for entry (e.g. a valid travel document).
Information on general rules of border crossing | A Magyar Rend?rség hivatalos honlapja
Read on the article for rules on air travel.
Accordingly,
"Alternatively, can I drive into Hungary from Serbia (US citizen) and then fly out of Budapest?" seems to be doable.