Just back up the posters above about the trails... There's a common, erroneous belief that they only consist of the 4 sections of the Sentiero Azzurro (Blue Path). 2 of those 4 sections have indeed been closed to landslide or rockfall for quite a number of years. I see that another section is temporarily closed to maintenance, which can happen to any of the trails at any time. The section between Vernazza and Corniglia is currently open.
But as noted above, there are MANY trails within the system - albeit many of them higher, longer and/or more strenuous - so unless the weather is too wet for safe hiking, you will find some open routes. Below are two link to the official park website:
http://www.parconazionale5terre.it/Eindex.php
http://www.parconazionale5terre.it/Esentieri-outdoor.php
As well, the villages themselves provide opportunities for walks/climbs to lovely scenic viewpoints, such as up to the Church Of San Francesco/ Capuchin Friars Monastery and nearby cemetery in Monterosso, and lofty cemetery in Vernazza. As you only have 1 full day in the region, you'll find more "pretty" to see than you can cover in that time. Oh, and it really is just a few minutes between villages by train. :O)