Saturn doesn't really have a surface on which you might walk, so the following is a bit fictitious... Anyway, its equatorial radius is about 60,268 kilometers (according to Wikipedia); that would make a circumference of about 380,000 kilometers. Divide that by any convenient speed, to figure out the time. For example, 5 kilometers/hour, or (and considering that you can't walk 24 hours a day) perhaps 40 or 50 kilometers/day.
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Mimas takes about 22.6 hours to complete one orbit around Saturn.
The sun does not orbit Saturn. Saturn orbits the sun.
It takes Saturn about 10.7 Earth hours to make one full rotation around its axis.
It takes Saturn equivelent to 29 years to complete a full turn around the sun and it takes Saturn 10.5 hours to complete a turn on its axis. (Therefore a year on Saturn is equivelent to 29 years on Earth and a day on Saturn is 10.5 hours)
It takes Saturn about 29.5 Earth years to orbit around the sun once.