I did what you propose to do. Rosie was around 3 years old when she came to live with us. She was a 'problem cat' - had issues with the toilet and she had been harmed by previous owners I imagine - we took her in to give her a stable home as she kept being 'given back' to the shelter
Listen to Geekbabe's advice - you do
not own the cat - nor are you its owner. The cat owns you. I'm serious - too many people say 'my cat'.. doesnt belong to you
I went along to our local shelter and spent some time with a few of the cats there. Rosie was really playing around to get a bit of attention. Cat's personalities can change over the years. Rosie seems to have calmed down from her frantic state... she used to be afraid of plastic bags, wouldnt let us pick her up etc.
It took me
two years to earn Rosie's trust - and even sometimes she isnt sure about a situation. Give the cat time. Rosie was naturally a loner - she would seek us out if she wanted company, or meow - if she didnt want company she would find a secluded spot and sit/lie down. Respect her space.
Rosie's toilet habits leave something to be desired. Turns out she was 'given back' by several famies who couldnt put up with it. Hey, my carpet in the front room is ruined, but I love my cat. Dad wanted to get rid of her but I stood in the way - I'm going to buy Dad a new catpet someday
Nowadays when I come home she can be all over me. I make a point of going to say hello but if she's asleep I wont disturb her too much. Might give her a biscuit or something.
My cat is my best friend.
Ever. Whenever I've had a bad day I sit down and within minutes the furry lump finds my lap and paws around. You cant stay ratty for more than a few minutes when faced with purring

, even after a hard day.
Its the best thing I've ever done. I will always give a home to cat(s).