Is their such a person that can be an Apostate and Atheist at the same time or is it a Contradiction?

An Apostate is someone who simply leaves their faith. It doesn't mean that they are necessarily without a religion. However, most Apostates become Atheists.

If I'm a theist of any kind and leave that religion, and can choose to be an Atheist in that same moment. Thus, Apostate and Atheist.

However, not all Atheists are Apostates. Some people are never a member of any religious group in their lives and never convert at any point. Thus, they are Atheist and that's that. They never had a religion to leave and therefore become Apostate.

Also, just because you become Apostate to one religion doesn't mean that you have to be an Atheist. You can convert from Islam to Christianity. Thus, an Apostate to Islam, but not a Atheist.

The word apostate may of course be applied in many regards. If, however, it is applied strictly by God (which usually doesn't happen) all who apostate from him shall be written down into the earth

That all atheists have apostated from God - is a given, perhaps. Perhaps, because Christ is the judge, not I.

Since churches apply it quite frequently to someone who leaves their denomination for another belief system, many atheists may be apostates from so and so church. The Wt people count anyone from their ranks who becomes e.g. Protestant, Catholic, LDS, an apostate. That naturally includes atheism. If other churches are that blatant about it, I do not know.

That they do not care about this - is also a given. On the other hand, atheists usually claim that believers in some deity simply belong to the great god-delusion. So, on each side there is a labeling taking place.

No. An apostate is a Christian who loses salvation. The Greek word is apostasia and it is in this Scripture:

2 Thessalonians 2:3 - Let no one deceive you by any means; for that Day (that’s the return of Christ) will not come unless the falling away comes first... ~ The falling away is not a so-called Pre-Trib rapture as some people think. The word for falling away also means to forsake, defection, apostasy. This is about apostate Christians who rebel, defect, and fall away from God. In the tribulation, when more severe persecution comes, many believers will grow cold and abandon the faith. Only those who continue until their end will be saved. Yes, a pre-trib rapture is also false doctrine. The safe way is the right way to believe. It’s better to be safe than sorry. If you believe pre-trib and you are wrong, then you might be so heart-broken when you find us in the tribulation, that you will fall away. So get strong in the Lord, b/c tough times are coming.

I guess you'd be an apostate first in the eyes of your religion if you became an atheist.

No, apostates are often atheists.

No contradiction.

I was about to say yes until I realized it's not "A prostate".

a·pos·tate - noun

One who has abandoned one's religious faith

I would think they go hand-in-hand in many cases.

no