Rajasthan’s desert-influenced climate brings genuine seasonal extremes, and Jaipur specifically gets very hot in summer (April to June, regularly exceeding 40 degrees) and can turn surprisingly cold on winter nights (December to January, occasionally dropping close to freezing) despite warm daytime temperatures. Layering matters considerably in winter, with a warm layer for evenings even though midday remains comfortably warm.
Rajasthan is culturally more conservative than many parts of India, and modest dress — shoulders and knees covered — is the sensible default for both men and women, not only at temples but in general public life, particularly outside the most touristed central areas. This is also practically sensible given the intensity of the sun: loose, lightweight, long-sleeved cotton clothing protects against sun exposure more effectively than minimal clothing.
At the City Palace, Amber Fort and the various temples around the city, headscarves or modest coverage may be specifically required in inner sanctums, and carrying a light scarf or shawl for this purpose is a practical solution that also doubles as sun protection.
Comfortable, sturdy walking shoes matter given the significant amount of walking and stair-climbing involved at Amber Fort and the City Palace, both of which cover substantial ground across uneven historic stonework.