We don't really know anything of them, or even if they ever existed; like the Others they are a legend in modern day Westeros. Apart from the quote you mention from Old Nan, other people have told similar tales of the ice spiders.
The white walkers of the wood, the cold shadows, the monsters of the tales that made him squeak and tremble as a boy, riding their giant ice-spiders, hungry for blood...
A Storm of Swords, Samwell I
The one that killed Small Paul was riding a dead horse, so that part's plainly true. Some accounts speak of giant ice spiders too. I don't know what those are.
A Feast for Crows, Samwell I
The tales go on to say they rode monstrous ice spiders and the horses of the dead, resurrected to serve them, just as they resurrected dead men to fight on their behalf.
The World of Ice and Fire, Ancient History: The Long Night
Other than that from Old Nan's tales we hear that they are as "big as hounds" which Bran later recounts.
All Bran could think of was Old Nan's story of the Others and the last hero, hounded through the white woods by dead men and spiders big as hounds.
A Game of Thrones, Bran IV
As for if there is any foreshadowing for this event, in the dream you mention that Jon has he mentions people climbing the ice like spiders.
Burning shafts hissed upward, trailing tongues of fire. Scarecrow brothers tumbled down, black cloaks ablaze. "Snow," an eagle cried, as foemen scuttled up the ice like spiders. Jon was armored in black ice, but his blade burned red in his fist. As the dead men reached the top of the Wall he sent them down to die again. He slew a greybeard and a beardless boy, a giant, a gaunt man with filed teeth, a girl with thick red hair. Too late he recognized Ygritte. She was gone as quick as she'd appeared.
A Dance with Dragons, Jon XII
This one is quite a leap and is probably just nothing but it could be a very subtle hint for the ice spiders.
"I had been given to understand that those castles were ruins. Dismal places, bleak and cold, hardly more than heaps of rubble. At Eastwatch we heard talk of rats and spiders."
The cold will have killed the spiders by now, thought Jon, and the rats may be a useful source of meat come winter. "All true, Your Grace … but even ruins offer some shelter. And the Wall will stand between them and the Others."
A Dance with Dragons, Jon XI