1) With the emphasis on 'bored,' the meaning is almost the same as saying "I am bored." We don't usually put emphasis on 'kind of' or 'sort of.' The emphasis is on 'bored.' You're either bored, or you're not.
2) With the emphasis on 'very,' this sentence has a stronger meaning than 'I am bored.' But in this case, we usually say, "I'm so bored" (with the emphasis on 'so'). Using 'very' in this situation is less common.
3) and 4) have the same meaning as 1). However, we use 'rather' more often than 'fairly' and we usually say the sentence a little slowly, to emphasise the meaning.
The intonation goes down on 'bored' in each sentence.