Anuradhapura or Polonnaruwa if I only have time for one ancient capital

If you can only pick one and want a satisfying single-day visit, choose Polonnaruwa. Here is why: Polonnaruwa was the capital from the 11th to 13th centuries, so it is more recent, more compact and far better preserved. You can cycle the whole core in a day, the sculpted Buddha images at Gal Vihara are the artistic highlight of all Sri Lankan ruins, and the layout is easy to follow without a guide. Anuradhapura is older (3rd century BC to 11th century AD) and arguably more important historically as a still-active pilgrimage site (the sacred Sri Maha Bodhi tree, the Ruwanwelisaya stupa), but it is sprawled over many kilometres of bush and feels overwhelming on a single day. So: Polonnaruwa for first-timers and visual reward; Anuradhapura when you can give it two unhurried days plus Mihintale next door.