What specific day were the lantern used? Did they have Chinese New year in the Han dynasty

Nowadays, they are widely used in Chinese New Year and Lantern Festival.
Yes, they have Chinese New Year in the Han Dynasty. Actually, it's in Han Dynasty, the Chinese New Year Day was set on a specific day (January 1st of Chinese lunar calendar). Before that, it was celebrated on a few different days by people in different regions of the country.