the most economical way is to use grass seed and use a seed spreader. there are hand held models or push from behind types. some can be rented but check the rental rates, occasionally the hand seeders arer more economical.
It is very rough terrain and muddy. Almost 1/4 acre. How do I flatten it first?
Spreading seed with the hand held model is probably the best way since a push type will be difficult over rough terrain. The turn the dirt to cover the seed or spread hay so birds won't eat the seed. Fertilize with 13-13-13. Stay off the area as much as possible. Depending on where you are and the type of grass, you may be too late for seed and need to opt for sod. If so, you can choose a spreading variety and lay in a checkerboard pattern allowing it to run and fill in over time.
That's a tough one. We have a tractor and use it to smooth out areas. You could have topsoil brought in and "level out".
If you don't have access to a tractor, seriously consider renting one or hiring someone to do it for you. It would be terribly back breaking to do with a shovel and rake.