1st of January is normally a holiday in Japan. However Takeshita Street is a popular tourist destination. Will stores be open on Takeshita Street on the 1st of January?
Some, if not most, stores are open.
Note that I haven't visited Harajuku on New Year's, so my information is second-hand, and most of my references are in Japanese.
Looking through Yahoo Q&As (like this and this), I'm getting mixed signals: some says a lot of shops are open, and others say most are closed (also note that some of the posts are old, so the situation may have changed).
A quick search tells me that food shops are open, and larger malls like Laforet Harajuku and Omotesando Hills are also open. I checked a few stores on the list (requires a good amount of digging) and most of them had regular or holiday hours. One or two were closed (probably looked up 7 or 8 stores).
Looking at this video of Takeshita in 2017 (taken around 15:00), I believe it's safe to say that a lot of the stores were open for business. Someone with first-hand experience would have a better say than me, though.
Also note that Harajuku station is one of the closest stops from Meiji Jingu Shrine, so it will be crowded on New Year's.