D., this was exactly how my older DD was at ages 2-3. She shoved. She hit. She pushed kids, sometimes unprovoked. She would sometimes go to the top of a slide and push the kid sitting at the top down the slide because she didn't want to wait her turn or share the slide. She took toys from smaller kids. Out of the blue, she once growled on the way to the doctor's waiting room and mumbled angrily "other kids" Come to find out she did not want to share any of the toys or crayons if there happened to be other kids there that day. I totally know how helpless and sad that makes you feel as the Mom. But please know that it really isn't bullying at this age. It was a phase. It is about learning empathy, impulse control, and sharing the world with others. And some little kids have a harder time with this.
For us, it was a phase, thank goodness. When I first noticed it I reacted much like you. Totally startled, rushing in to correct it immediately, apologies, long lectures, stern approach. I felt horribly guilty and compelled to nip it in the bud. However, my DD noticed it got a lot of attention, and it only make the issue worse for awhile. She would try to start conversations with Grandma saying "I hit kids" because she knew that would get her HUGE attention and long conversation about why she shouldn't act this way. I had to train Grandma to ignore this talk. Wasn't easy. See of course it alarmed Grandma too, and she felt compelled to rush in give it loads of attention to fix it.
My DD stopped her pushing/hitting sometime in her 3 year old year. Towards the end of that kind of behavior she pushed someone down the slide at the playground. I picked her up and left. Immediately. No huge lecture. No second chances. Swift consequence. You push, we leave. I believe it made an impression. Never happened again.
Also preschool helped trememdously. She had a scowl on her face one day when I picked her up. I asked. What's wrong? She said, when I did this (she made a shoving gesture) Johnny pushed me back and it hurt. I said yes, "pushing makes people angry, and if you push you will often get pushed back much harder. And it will surely hurt." Never happened again.
I would talk to his preschool teachers since you've heard its happened at school and let their professional experience guide the situation.