Yes, they do, kinda. From
http://www.amonline.net.au/spiders/resources/biology.htm :
"Do spiders sleep?
It really depends on how you define "sleep". All animals have some sort of 'circadian' rhythm - a daily activity/inactivity pattern. Some are active during the day - diurnal - others are active at night time - nocturnal/crepuscular. The periods of inactivity are characterised by withdrawal (to a shelter perhaps) and a drop in metabolic rate.
This applies to spiders as well, although no studies have been done to measure the period of time spent in such a state or at what times different species do it. It seems that spiders with good eyesight that rely on vision to capture prey may tend to be more active in daylight hours, whereas others that rely on snares/webs could be active at other times, but this is not necessarily the case for all species.
In cold climates, spiders "overwinter", which means that they have a kind of hibernation period. Overwintering involves a drop in metabolic rate, where the spiders bring their legs into their body and remain huddled in a shelter during the coldest months of the year.
This ability to shut down for a long period of time indicates that they might be able to do it for shorter periods in their everyday cycle, which could be seen as a form of sleep or rest.
Information from Foelix, R.F. 1996. Biology of Spiders. Oxford Thieme. Arachnology section, Australian Museum"