These TCG rulings were issued by Upper Deck Entertainment and have since been deemed unofficial by Konami. They were previously considered official and can still be correct, as long as they follow the official gameplay rules, or unless Konami has issued a ruling that says otherwise. References: 
"Asura Priest" may attack directly if it has not already attacked a monster this turn.
If "Asura Priest" attacks a monster and your opponent activates "Magic Cylinder" to negate the attack, "Asura Priest" can attack that same monster again.
If your opponent has "Patrician of Darkness" on the field they can choose the same monster as the attack target of “Asura Priest” each time you declare an attack with it. (For example, re-direct all of your “Asura Priest” attacks at “Spirit Reaper” instead.)
After "Asura Priest" has attacked monsters, other monsters you control may still attack directly.
"Asura Priest" can attack each monster once during the same Battle Phase. These attacks do not have to be all in a row. For example, you can attack monsters with "Asura Priest", then attack with another monster, and "Asura Priest" can still attack again as long as it attacks a monster it has not attacked yet.
These TCG rulings were issued by Upper Deck Entertainment and have since been deemed unofficial by Konami. They were previously considered official, but may or may not necessarily hold true nowadays.