Shinjuku assassin

Not yet. I believe he has a rank up on the way but at the moment he is pretty bad. Gamepress placed him in tier 5 and I think I agree with them.

Short answer is no, as mentioned he gets a buff eventually but it won't be until Halloween 2020. He is still an SR though and thus generally worth using over comparable servants of lower rarity such as Fumma or CA Hassan(though worth noting both have better defensive options than him which can matter in some harder fights)

No, he is one of the worst 4* in the game. He belong to the trash tier that will inevitable find its way to your chaldea when you are rolling for something that it is actually good... Like fionn, steno and loli medusa.

4* Jing Ke, Kotarou and CA Hassan do his job (powerful active stargen, decent crit) better, while hitting far weakly.

On October 2020 he got a Quick up and became good for quick crit.

The answer is yes and no, he is hard to use and has a very small niche. He is centered towards controlling star distribution by raising/lowering his own absorption while generating stars. He is not the best star genrerator (Fuuma is better than him) but if you're using a berserker and only he and Assassin have cards, you can gain 20 stars and make (almost) sure the berserker gets all stars.
Unless you really want to cover this 1 niche, don't use him, he is almost worse than Stheno with his only advantages being his niche and NP.

I agree with other peoples' opinion and gamepress's evaluation/analysis. What he looks like just a SR servants with similar or even worse performance/function as Kotarou, Jing Ke and Cursed Arm. 3 Quick card recently is a kind of burden (except you have Jack's great NP gain and Quick card hit accounts).
You can expect his boosting performance after 2nd skill upgraded and team with Skadi, but right now use him is really a luxury choice. In other word, you have to upgrade all his 3 skills to lv 10, but still meet challenge for other considerations, like poor survivability/durability, mediocre base attack status, almost 0 resistance to debuffs, etc.