Hi Christa -
If I start in Montessoro and hike to Vernazza, how long is the hike
It's 2.27 miles (3.668 km). You could also go on from Vernazza to Corniglia, which would add another 2.5 miles (4.137 km) or do it the next day. The Monterosso>Vernazza route is quite steep in places and involves a lot of steps so make the decision based on ow much energy you have left when you reach Vernazza.
do I need a hiking pass
Yes, both of these trails are part of the larger, 4-segment Sentiero Azzurro, or "Blue Trail". This is the most popular route in the park, and these are currently the only two segments which are open: the other two are under repair. You do need a park pass to hike any section of the "Blue". Both of these trails will be VERY busy.
http://www.parconazionale5terre.it/page.php?id=157
There is also a more expensive version of this card which covers train journeys within the CT:
http://www.parconazionale5terre.it/page.php?id=158
how do I get to Montessoro to start the hike?
By train. Either buy single tickets or the Cinque Terre Treno MS combo park pass/train card. If you buy individual tickets, you must validate them before boarding the trains. If using the pass, you only validate it the first time you use it. Fines are high if caught with unvalidated tickets/passes so don't forget! You may also return by boat (not covered under the park/train pass) but I'd save that for a longer journey along the coast: say, from Vernazza to Riomaggiore and then train back.
If you hike from Vernazza to Corniglia, you'll also take a train back to Vernazza. Taking a boat isn't an option from that one.