Ceiling fan and light work with pull chain only, not light switch
It looks like you have a switch loop so you need to take your always hot black wire and connect it to the white wire with the black markings (can you read the markings). Put some black tape on that white wire to show it's always hot. Hook your fan up to the remaining two wires. Put some black tape on the white wire at the switch box too. Leave the ground from the fan unhooked since you don't have  ground there.
Now for the bad news. I doubt that that ceiling box it rated for a ceiling fan and it needs to be changed out. While the box could be firmly mounted to the joists, the threads and screws you use to attach the fan to the box are not strong enough. If you have attic access, changing out the box won't be too bad but if you don't, You might have to remove some of the ceiling to replace the box.