Rental house-UGLY exterior doors!
Get some wood veneer, trim it to the size of the doors, use adhesive to attach it. Then put on a good quality outdoor varnish.
First of all, be sure the owner of the property will let you do whatever you want.
I get the feeling that as long as he is not paying for it, it's not a problem.
Take a photo of it first and then when you pay the rent, get a written OK. It may come in handy when it comes time to get your deposit back!
I like the idea of covering the door if you can't paint it to look good.
I would not spend alot of money fixing this place up. The owner is not interested in fixing it up or he would have. I'm sure there are going to be more surprises for you down the road, don't get him in the habit of you paying to fix the place.