Reading the One Punch Man wikia I've read that when a younger Saitama, before being a hero, was able to defeat a monster. How was this possible? Did Saitama have some power before becoming a hero?
In the anime and manga, Saitama starts off as a reasonably fit, unemployed salaryman (he's notably more non-descript in the original webcomic). He's not as cut as he is portrayed in the present time, but certainly not doughy or without definition.
Saitama's fight with Crabante, his first monster fight, is indicative of the mindset that would deliver him unto extreme power: don't stop until it's done, and ignore your limits. He fights through pain and Crabante's superior power to take advantage of Crabante's weak point and saves the day.
In the manga there is a sidestory chapter (Volume 2, Bonus chapter) that takes place around 300 days into his training to be a hero. He still has his full head of hair. He's in a great deal of pain through the initial part of the story, but near the end an attacking monster hits him while he's out doing his running and knocks out his tooth
which turns out to be the source of Saitama's pain, as he had an awful toothache in exactly that tooth.
This restores Saitama's focus (and he suffers no other form of damage, despite being launched through a wall), and he quickly one-shots the monster and simply continues on with his running like it was no big deal.
Indeed, there's a second sidestory chapter that takes place during his training days (Volume 3, Bonus Chapter 1). This one I don't think specifies exactly when it occurs, but he still has his hair. Saitama's out in the city and really, really needs to pee. But he can't get inside anywhere to use a bathroom because everyone's hiding from a huge attacking monster (Cicada Nymph). Saitama runs pee-dance-style trying to find one and one-shots the monster when it gets in his way. I don't think Saitama really even registered the monster (either as a threat or even existing), being more focused on finding somewhere to pee.
Which is all to say that Saitama does appear to have a meteoric rise in power, but in his fight against Crabante his only extraordinary ability appeared to be his willpower and courage. He does not achieve his ultimate power until he loses all of his hair, but even before he has shown any hair loss (it's not specified if he lost it somewhat gradually, or all-at-once, or what) he is able to take powerful hits with little to no damage, and can one-shot monsters.
Incidentally, the bullying thing you mentioned is in the Volume 1 bonus chapter.
I believe if you follow the story, we can see that Saitama was not bald headed. He was a normal salary man. Then one day he met with a crab monster, where he was beaten so badly. With some luck and brain, he managed to defeat the crab monster. Later on he fight with monster and train a lot. Just like what anime season 1 stated, he do some muscle training to his limit until he realized that he has become bald.
So imo, he was a weakling before becoming a hero. He was strong before joining hero association.