Cost really is regional. Landscapers usually will give you estimates. A lot depends on the cost of hourly labor and if you are putting in sprinklers (Grass needs a lot of water). It also depends on the amount of amending you have to do and the kind of grass and if you are seeding, plugging or sodding. I think 200 pounds might be a little low. it would probably not even cover some of the material costs.
If the ground is not rock hard, and you are willing to do this in sections, you could work the amendments in (compost, soil conditioners and fertilizer) with a shovel by double digging. It is slow but actually causes less damage to the soil than a tiller. If the ground is uneven and you want to re-grade you need to make sure that water still drains away from the house. 5mx5m is not all that big so it can be done by hand. But easier if you host a weed party and get a few friends over to tackle the job. You can go over to a nursery that sells compost and amendments or turf, ask for advice on what to use and how much you will need or do your research first so you know what you need to do and what it will cost. You will also need to get enough tools to do the work and that means good shovels, rakes, and gloves. You will get better advice where the nursery is their main business, not a hardware store. Here there are a couple of turf farms and one of them even has instructions for how to prepare the site and install the grass and the after care. It helps if you have one or two people who have experience. It is easier to grade when you do the whole thing at one time. Seeding is the least expensive and take the longest to establish and sodding would give you an instant lawn but be the most expensive option.
Seeded grasses and stolons grow evenly but you would need to water frequently and it will look like a weedy mud hole most of time. You also need to limit traffic as much as possible to minimize the pits. If you don't have a sprinkler. You can use a multi function faucet timer and hose attached to a yard sprinkler and set it to water up to 4 times a day for however long it takes for the water to get down about 4-6 inches. You will need to keep that watering routine up until the grass is established before you can back off watering. Work only a section large enough for your sprinkler to cover otherwise you will need to move the sprinkler and turn the faucet on and off manually up to 4 times a day depending on the season and how fast the soil dries.
Grass is very high maintenance, you will still have to weed, fertilize, water and mow forever after that, although the time in the beginning will be the most time consuming.