Fill and pack the holes/area in question to about 1/2" of the 'top' of the surrounding topsoil. Fill a few inches at a time and TAMP. This way, you'll do a good job of filling in the air holes. If you try to tamp too much dirt, you'll have airpockets that will eventually collapse later and you'll have depressions in your new lawn. Use a 2x4 or a 4x4 for the tight areas. Use a TAMPER for the bulk of the work.
Then, fill in 1/4" of the last 1/2" of the hole and tamp. Lay in the PROPER grass seed. If the 'spot' gets lots of sun, use a grass for that purpose. If it gets lots of shade, use a grass for that purpose. And so on. You shouldn't have to prepare the soil any unless it's very acidic (like you use to have pine trees there or something.) Then, cover the remaining 1/4" of the hole with topsoil but DON'T TAMP. Water daily in the morning before 9am until the grass is firmly established. Keep everyone off it until it's established. Use native grass(s).
If you need more info, ask away. My best to ya and hope this helps.