Is SpaceX lunar mission only a flyby mission, or it is going to land on Moon too?
Pretty much certain it's a flyby. Grey Dragon mission was a flyby. To send a mission to the surface of the Moon would require many refueling missions, and I can't imagine that would be the case.
Also, the tweet that started everything says so.
SpaceX has signed the world’s first private passenger to fly around the Moon aboard our BFR launch vehicle
Update: The mission will be a flyby of about 4-5 days of a mission carrying a dozen artists who look upto the moon as an inspiration and wish to create an artwork that's global. A Japanese collector and billionaire fashion entrepreneur Yusaku Maezawa wishes to propel the BFR rocket by funding it and securing tickets on the first private flight to the moon by about 2023.More information can be found at articles from various news outlets such as this https://www.cbsnews.com/news/spacex-moon-announcement-elon-musk-reveals-bfr-moon-mission-passenger-yusaku-maezawa-2018-09-18/