Avenger of Shinjuku, like all Avengers, has a high attack value, but unfortunately his skills and his NP are not that good, besides you will hardly see battles in which you have a Ruler as an enemy.
Fallen Demon allows you to absorb stars during a turn (which will be good for you, because as an Avenger it will hardly give critical blows). for 1 turn. This, combined with Monstruous Strength, gives you a 10-30% buff for 2 turns. Finally, One Cloaked in Death will serve you, above all, to remove buffs to a specific enemy, such as evasion, defense, critical damage, etc. and to kill any non-boss enemy with your NP.
All three abilities, at nv10, have a cooldown of 5, but their use only lasts you one turn at most (2 turns that of Monstruous Strength). That means that, apart from always hitting neutral for his class, you will only do decent damage during one turn, and only if you have prepared some quick attack to generate stars in the previous turn.
His NP is Quick and seems to hit a lot, but nothing could be further from the truth. Even with Skadi, the damage is pretty mediocre and doesn’t fill your bar like it does, for example, Valkyrie or Parvati.
That said, he is a servant who does not support the team beyond removing a debuff and also does not do much damage by almost always doing neutral damage and being a quick servant (since none of his skills increase critical damage).
And finally, to increase their skills they ask you for a lot of materials, not gems, so it’s even harder to take advantage of it.
I have it at 70, being my only avenger, in 4-4-4 and beyond bosses rulers, I don’t use it. Only train it if you don’t have any more SR or SSR servants or if you like the Wolf design.
Having Waver, who is one of the best casters in the game (if not the best, since his skills work for all kinds of teams and strategies), I suppose you could give him a try, since one of his skills increases the damage I criticize, if I’m not mistaken, in addition to giving a 30% Np load. But keep in mind that you will only use it against Ruler bosses or if you face a boss and you don’t have many Servants with an advantage against him. (I don’t know how long you’ve been playing, so if you’ve started the game recently, it’s not a bad “neutral” unit). Next month they will give Mecha Eli-Chan, an alter ego booster that you will have at NP5 if you do the event, very good against cavalry units and bersekers. You can reserve materials and experience orbs for her if you are not convinced by wolf.
I hope I have been helpful, I am Spanish so if there is something that is not well written, excuse me.