When excavating the Grail, the hero loses their entire turn. It is not allowed to excavate the Grail on the day of arrival at the location of future excavations. However: There is one caveat that is quite thoroughly discussed in one of the topics on HeroesPortal - "A puzzle from Arkh." The essence of the caveat is that there is a method that allows you to excavate the Grail on the day of arrival by varying the hero's maximum MP level. Details can be found at http://www.heroesportal.net/tavern/?id=150632.
I read about this trick in the physics section, went to the forum, and found out that someone suggests transferring a slower unit to the hero before excavating the Grail, and this should allow you to excavate the Grail even if you don't have a full turn. However, despite all my attempts, I couldn't make it work in SoD. Maybe I misunderstood something?