I could hardly believe it but I found their tallest player was shorter than I.
Generally in sentences with inversion the pair of conjunction with hardly is when e.g. Hardly had I arrived home when my phone rang. So going by this isn't the but in above sentence incorrect and its better to use when here or it's an exception here ?
The word hardly is being used in a different sense in your two examples. In the first, it qualifies your ability to believe something. In the second, it states a fact, meaning "just as I arrived home, ..."
Using when is correct
I could hardly believe it when I found their tallest player was shorter than I.
Although than I is formally correct, than me is more usual, so I would modify the sentence in steps:
I could hardly believe it when I found their tallest player was shorter than I am.
I could hardly believe it when I found their tallest player was shorter than me.
I could hardly believe it when I found I am taller than their tallest player.
Using shorter makes the sentence harder to understand, since it is about being tall, not being short.
Back to the question, but is used to contradict something, or give an opposing view. There is none here to challenge. Perhaps if you said
I believed their players were very tall, but I found I am taller than their tallest player.
the but would be well placed.
Hardly had we come in the door... when the phone rang.
Hardly had we taken our seats when the curtain opened and the play began.
I could hardly believe my eyes ... I was taller than their tallest player.
The sentence could be correct with either the word "when" or the word "but", though the emphasis would be different depending on which word is used.
If the sentence uses the word "when" then the second half of the sentence would be simply giving the situational context to the the first half of the sentence. That is to say, it would be describing the situation in which you could hardly believe. The situation was when you found out that their tallest player was shorter than you.
If the sentence uses the word "but" then the second half of the sentence becomes a contrast to the first half of the sentence. The first half is saying that something was unbelievable, while the second half is saying that something is true. Thus the complete sentence is saying that something is true despite the fact that it is unbelievable. In this case it is unbelievable that their tallest player could be shorter than you, but it is still true as you have found out.
