Erratic behavior (new cat owner)

So last month I got a cat, it was my neighbors cat and he travels a lot so i use to take care of her during weeks he was away. He told me last month he was planning on giving her away because he simply couldn’t take care of her because he’ll be traveling the rest of the year. I told him I’d gladly take her so I’ve known her and already had that bond. She’s 2 and a half years old, pretty sure she’s had her shots but I know that she isn’t spade. She’s so sweet but I’m curious what her behaviors mean and if there are any underlying issues or anything I’m doing wrong. So her behaviors differ from week to week, one week she’ll just chill and sleep all day, and the next week she’ll be crazy running around biting my feet etc. shes well fed, has a clean litter box, and lots of toys. Is this normal cat behavior or does she get these random spurts of energy for a week because of hormones because she isn’t spade. I know I need to get this done and I’ve only had her for a month, so curious what I’ll need to do and how to go about it.