His friends would probably take care of him. Midoriya would be very eager to help, using his kind-hearted nature. He would make sure Todoroki is fed and comfortable.
One idea could be that all his friends at UA plan a surprise birthday party for him. They decorate the common room with his favorite colors, blue and red. Another plot could be Todoroki spending his birthday with his family, trying to reconcile with his father on this special day. Maybe he receives a heartfelt gift from his mother that has a deep emotional significance.
Todoroki's character development in time travel fanfictions can be quite profound. He may start as a somewhat aloof character, but through time travel, he begins to understand the importance of relationships more deeply. When he travels back and witnesses the relationships he could have had or lost, it makes him cherish the ones he has in his original timeline. His view of heroism also evolves. He realizes that being a hero isn't just about using his powers but also about making the right choices at the right time, which is especially highlighted when he has to make decisions that could change entire timelines.
In the fanfiction, their relationship could start with a bit of competition. Both are ambitious to be heroes. However, they soon realize that they can achieve more by working together. As kids, they might compete in small challenges like who can run faster or jump higher using their quirks (or in Izuku's case, his physical training). This healthy competition could fuel their growth and transform into a deep - rooted friendship that withstands all the difficulties they face later in their hero careers.
One common theme is family. Given Todoroki's complex family situation in the original story, many fanfics explore his relationship with his father, mother, and siblings further. Another theme is friendship. They often show how Todoroki forms deeper bonds with his classmates.