In your question you ask which of the below sentences are right.
1)I can't take part in the today's activity because I'm A Night Shift today.
2)I can't take part in the today's activity because I'm Night Shift today.
3)I can't take part in the today's activity because I'm The Night Shift today.
The answer is that NONE of the above sentences are correct. There needs to be the preposition "on". In addition, the reference to the time of the day needs to be "tonight" as that is when the activity will take place. The example sentenc is as follows:
I can't take part in today's activity because I'm of night shift tonight.