Anime with a prince and a servant chased by zombies who meet an immortal woman
It is Kurozuka:


  The series begins in 12th century Japan and centers on Kuro, a character based loosely on the legendary Japanese swordsman Minamoto no Yoshitsune. Kuro and his servant, Benkei, meet a beautiful and mysterious woman named Kuromitsu while on the run from Kuro's elder brother, who seeks his life. Kuromitsu and Kuro fall in love, but he soon discovers that she harbors a terrible secret: she is a vampiric immortal. Following an attack by his pursuers, Kuro is badly injured and must imbibe Kuromitsu's blood to save his own life. Kuro is then betrayed and attacked by Benkei, who has been subverted by a shadowy organization called the Red Army, and Kuro's head is severed, which interferes with his transformation into a fully immortal being. [...]