An upgrade from 7.7.1 to 7.8.2 should be relatively straight forward. You would need to upgrade the software on each of your servers - Web Server/App Server and any machines that have the console installed. Once you have upgraded the software, you would need to run the database upgrade from within the configuration center web application. Prior to running the upgrade, a database backup would be highly recommended. As for how long it would take, I cannot say, but as the version hop is relatively small, I would say a couple of hours plus.
Some useful information can be found here: Announcing LANDESK Service Desk 7.8.2 and here LANDESK Service Desk 7.8.2 - What’s New in LANDESK Service Desk. More in depth information about the upgrade can be found here: LANDESK Service Desk 7.8.2 - Upgrade Guide
If you feel totally uncomfortable about the upgrade, please do not hesitate to contact your account manager to discuss a Professional Services Consultant visiting you.