The word 蜘【くも】蛛 consists of two kanji, 蜘【くも】 and 蛛, but in only 蜘【くも】 has the furigana くも.
Does this make the 蜘 part silent?
If this is the case, are there rules for silent kanji?
No, the reading くも is a special reading assigned to the entire compound 蜘蛛, not to the first kanji. You might see the furigana for the compound printed over the first kanji sometimes, depending on how it's aligned, but if so you should interpret it as the reading for the entire compound.