If I purchase app. Is it for the month or the year?

Typically, when you buy an App, you pay one price and that's for the life of the app. In other words, if an app is $4.99, you pay $4.99 once and that's it. Now if you subscribe to app, such as Apple Music, you pay a monthly subscription fee.

What app are referring to?

I just bought a Samsung Smart TV and it was a 2018 model. I had no problem downloading the Apple TV app. I think that I had to update the software first. I did it about 6 weeks ago so I’m a little fuzzy on it but did you try updating the software on the TV yet?

You should also be able to use AirPlay on the TV without needing a third party app. I was able to mirror right out of the box. Did you try to mirror anything at all?

Did you check with Samsung to see if the TV supports the Apple TV app? I purchased a Q70 series device for Christmas. I don't have an Apple TV subscription, but I just went out and checked the TV and was able to search for and select the Apple TV app. Exactly which version Smart TV do you have?

Looking at this support document from Apple, I don't see your device listed. Apple TV app - Devices - Apple

My 2019 Samsung Smart TV (Quantum Dot 65" Q 70 Series) came with AppleTV+ and Apple Mirroring 2 built in. I see Chris checked and your TV does not include AppleTV+.