I can't say that getting tested was fun or a cultural experience because it's always nerve wrecking. But, I really liked the service the Shinjuku health center provided.
I knew they offered service in English, and as soon as I arrived somebody asked me if I would like to be counseled in English. A few minutes later a lady who spoke flawless English met me in a private room and gave me a small talk on STDs. She asked a few questions about my sexual habits and then explained the procedures. One of the things that I was glad to hear was an explanation of how my health insurance would work in case I tested positive for anything.
In the end I got tested for everything they offer, which is chlamydia, syphilis and HIV. I had to give a urine and blood sample, and the person who spoke English stayed with me the whole time to make sure I understood the rest of the staff and what was happening.
I was then told to go back in one week to pick up my results. The same lady was there to translate them, and I was happy to hear that it all came out negative.
Everything was free and anonymous. I was given a number to identify my samples, and asked to provide a four-letter combination in case there was further need for confirmation. Otherwise, nobody ever asked me for my name, nationality, legal status in Japan or anything that could identify me in any way.
You can get these services at a private clinic, especially if you need the results faster, but there's usually a fee involved. I was really happy to go for a confidential and free test in Shinjuku, and can only recommend their services if you think you need to get tested.