What Is the best drought resistant small green ground Cover that stays green year Around? I don't want to mow grass if I Buy a property. Ok?

Couldn't say without knowing what climate you're in.  And how small you want it. And how much traffic it will get.

If it is a sunny location, then sedum is a great alternative to growing grass.

I have a sunny slope that is dangerous to mow. I bought what is called a "sedum tile" at either Lowes or Home Depot in a 12" square tray (I noticed the 'tiles' are smaller this year) . Yiu cut them up, like a tray of brownies and plant them 12" apart. In no time, they grow together and form a dense mat so nothing else will grow.

In my shaded areas, I grow yellow arch angel. This ground cover can be invasive, but that is exactly why I planted it because I do not need to mow where it grows.

Not knowing the property, the zone, the sun exposure or the soil, it is hard to say what ground cover will work best. 

Once you have your property, you can Google ground covers with your specific requirements and get the perfect ground cover for your area.

When I decided to plant ground cover the following growing season, I purposely left the leaves in my yard over the wintertime until the following May. This killed what was growing in my yard, so when raked , I had the perfect planting conditions without having to pull up any growth of any kind. Perfect conditions for ground cover planting.

The only sure answer is Astro Turf

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If it has to be drought resistant then it will not stay green all year. 

Ask at a local plant nursery. NOT lowes or HD.

Couldn't say without knowing what climate you're in.  And how small you want it. And how much traffic it will get.

It depends upon where you live and the minimum temperatures, how much rain, etc.   Grass is always better.  Go around to commercial areas and look at the ground covers growing in the flower beds around buildings.  That is what you you are looking for.