TwinBee is one of Konami's best shoot 'em ups, and there's a certain joy to be had by shooting bells (which act as power-ups) and changing their effect. Arcade Archives: TwinBee is the only way you can play this classic on PlayStation 4, though, and sadly it hasn't made its way stateside. So you'll have to buy some yen and purchase it from the Japanese store.
If you're a fan of rugby, then you'll be glad to know that there is finally a great rugby games for consoles. The only caveat is that the game, Rugby League Live 3, never saw a release beyond Europe. It's worth importing though if you want a good rugby title since you really don't have other options.
Sega has done a good job at bringing over most of the Yakuza series as of late. There is one huge omission, though, and that's Yakuza Ishin. Set in Japan's Bakumatsu period, the game features past versions of Yakuza characters and is a lot of fun. There's even an excellent translation available to help you get through it if you happen to get stuck (although Yakuza veterans shouldn't have a hard time).
Blade Arcus from Shining EX is a solid 2D fighting game featuring characters from SEGA's Shining series. It's a four button fighting game with three strengths of attack, and a button dedicated to using support abilities. It's very import friendly if you know your way around a fight stick.
While Konami's Power Pros series has come stateside in the past, we're not so sure about Jikkyou Powerful Pro Baseball 2016. That said, it's the very first game in the long running series to release on a high definition console. So if you're looking for a nostalgia play, or just want an alternative to MLB The Show, then Konami has you covered.
Puyo Puyo Tetris is one of the best puzzle games released in years, and it won't be coming outside of Japan due to licensing issues. That sucks. Thankfully, it's fairly import friendly and once you get the hang of things you are in for a blast. If you import one title on this list, make it Puyo Puyo Tetris.
Who doesn't love sudoku? The brain teasers are a great way to keep your mind awake and active, and Puzzle by Nikoli 4: Sudoku offers up plenty of them on your PS4. There's practically no barrier to entry here import-wise, and you're in for some fun puzzle solving.
Dengeki Bunko: Fighting Climax Ignition might end up getting a western release, but so far the upgraded version of Dengeki Bunko: Fighting Climax is Japan-only. It's a pretty fun mashup of SEGA and anime characters, and it's probably the closest we'll see to a Black Bullet action game.
Super Robot Wars OG: The Moon Dwellers is the dream RPG for mecha fans. It brings together mechs from several series (including Gundam), and for the first time ever the series will be available in English due to the Asian release of the game later this year. So make sure you import the Asian release, not the Japanese version, if you want to play in English.
Wonder Boy rules. Westone's classic series might be better known as Adventure Island stateside (which was an adaptation by Hudson Soft). If you want to play it on a PS4 then you'll have to get Arcade Archives: Wonder Boy off of the Japanese store. It's worth it, though, as this classic still holds up well.