Where do you live and describe what "soil is too bad" means...

I am in (Tampa) Florida...
I mean it's totally spoiled and seems like no one has touched it for years.
It has fungus and rust problem at big scale

I don't think you need to dig anything in. I would cover it with as much compost as I could afford along with some organic fertilizer at 10 pounds per 1,000 square feet. Then if it does not rain right away, water it for about an hour. If it does not rain in the next 2 weeks, then water again for an hour.

By putting the compost and fertilizer on top the soil and compost microbes will transfer the nutrients deeper into the soil by natural means. Unless that soil has been poisoned with chemicals, it should recover in time to put sod in for summer.

Is the area sunny or shady? I am thinking St Augustine sod would go down quickly and give you a nice green space immediately...after you have fixed the soil. St Augustine can stand more shade than the other Southern grasses, so if shade is in the picture, then your choices are limited to that. If you do the compost/fertilizer now you should be ready to sod in a month. You can fertilize again with anything you like after you have mowed the St Augustine grass 2 times. By waiting to fertilize you are ensuring the roots are down and growing. If the roots are not producing healthy grass growth, then you would be wasting fertilizer to do it sooner.