Is it that most of big shopping centre like takashimaya do not open on 1st Jan?
They open on 2nd or 3rd Jan?
The majority of shops including department stores are closed on Jan 1. Some open on the 2nd, but others not until the 3rd. Lots of restaurants are closed on the 1st as well, but you're not going to starve as some chain restaurants will be open. Konbinis are always open.
https://www.japan-guide.com/e/e3011.html says that Takashimaya is closed on Jan. 1, except for restaurants (department stores have a good range of restaurants on the top floor. You can look up other places in Japan Guide and click on Hours and Fees to see their New Year hours. Also see https://www.japan-guide.com/e/e2276.html
You will find most larger/chain stores open on the 2nd as that's the first day of the Winter sale. Although can be busy it can also be fun especially the lucky bags many sell. https://en.wikipedia.org/wiki/Fukubukuro
Some stores and restaurants will be closed. It might be noticeable, but there will still be many stores and restaurants open on the day.
For some places, 1st January is hugely busy as they sell Fukubukuro (Lucky Bags). I remeber going to Sunshine City on the 1st and there were massive queues (and the whole mall was open).