Welcome to Workshop. We're really pleased to have you join us and trust you'll get loads of helpful information, advice and inspiration for your home renovation work. Congrats on the new house purchase, we're looking forward to reading more about it.
Looks like you've got older-style plasterboard with hair in it. The good news is that the repair job is exactly the same as if you had newer-style Gyprock walls. The Bunnings D.I.Y. Advice How to fix a hole in a plaster wall has step-by-step instructions to follow. I'll also attach the video below and you'll see that they are repairing walls just like yours.
If there's a lot of damage to the wall, you might consider replacing it, but I'm sure you will find the repairs are quick and easy. In the Workshop discussion Patching holes in the wall, member @PartyDad had never done these kind of repairs before and got a great result.
Please let us know if you have any further questions after checking out the resources linked above.