Yes, a priest can get the Crimson Deathcharger mount in Wrath of the Lich King Classic, but not by personally completing the Shadowmourne quest line (which is only accessible to Death Knights, Paladins, and Warriors). The mount is a reward from the Shadowmourne quest chain and drops in the Invincible's Cache, but since this quest is class-restricted, priests cannot obtain it directly. However, the Crimson Deathcharger mount item is Bind-on-Use and can be traded, mailed, or purchased from other players who have completed the quest. Therefore, a priest can obtain the Crimson Deathcharger by acquiring it from another character or player on the account or server and then use it freely once learned, as the mount itself is not class-restricted. The only requirement is that the character has the appropriate riding skill to use the mount. In summary, priests cannot do the quest to earn the mount themselves but can still have and ride the Crimson Deathcharger if it is obtained through trading or account-wide means in Wrath Classic.