You can tap any untapped Spider you control, including one you haven't controlled continuously since the beginning of your most recent turn, to pay the cost of Arachnus Spinner's activated ability. You can even tap Arachnus Spinner itself to pay this cost.

You may search both your graveyard and your library, but you can find a maximum of one card named Arachnus Web.

If you choose to search your library this way, you don't have to find a card named Arachnus Web, even if one is there. You'll still shuffle your library after searching it.