Okay, I did a lot of Googling using various search terms, all with "spider" and including one of "Jesus", "Christian", "Christianity", "Catholic", or "Orthodox" and one of "myth", "folklore", "folktale", or "legend". This is what I've found.
Web over the mouth of a cave
There are several versions of this one, but in every case, one party is fleeing from another and hides in a cave. A spider spins a web over the mouth of the cave, so the searching party assumes that their quarry is not inside because otherwise the web would have been broken.
Web over Jesus' cradle
This is where a spider spun a web over Jesus' cradle/manger and it protected Him from all harm.
I only found two sources with this: spiderzrule.com and The Encyclopaedia of Superstitions, Folklore, and the Occult Sciences of the World Volume 2 (twice).
The Christmas spider(s)
In this German/Ukrainian legend, spiders create cobwebs all over a Christmas tree and then something or someone changes the threads into gold and/or silver. This is believed to be the source of the traditions of hanging tinsel on the tree and putting a spider ornament on it.
Spider on the cross
Unfortunately, I did not find any source with a story like this. In fact, the top Google results for "spider on the cross crucifixion" are almost all from this very website, Christianity Stack Exchange. If the tale does exist, I guess there isn't a record of it online. In that case, this question and answer is now the only record of it online!
So the answer to your question is (probably) nobody knows! If anyone does, they haven't said as much online (or in a book or article indexed by Google).