I recommend coming to Sensoji Temple in Asakusa at least twice: in the afternoon and in the evening. Due to the merchant street it is probably the busiest temple in Tokyo. At night you can take a relaxed stroll over the temple area and admire the illuminated buildings of the temple.
The temple is an open area, so there is no closing date. I could see a few tourists as well some locals there. Some preferred to pray in the evening since Senso-ji is not actually a very quiet place in the morning.
Actually I have been to Senso-ji at night on the first day of my very first trip to Japan. At that time I was just exploring the area and for some reason I ended up at Senso-ji. I used a different camera at that time (which I bought for the trip) and I didn’t bring a tripod with me so all pictures were blurred. Asakusa and Senso-ji are also featured in the DreamCast game Metropolis Street Racer. Unlike in the game, Senso-ji is not open for car traffic.
One night I saw a man with four dachshunds on the temple ground. The dogs were running all around and looked like hyper-active fur balls sniffing everywhere.