I'm new to dramas and have watched some of the best I've seen. A few of them are romance, since, some of them are not my thing.  

Would also like recommendations on classical good romance dramas (Planning to watch Proposal Daisakusen)

About my topic, I haven't seen that many of "Out of Their League" romance.  So, yeah.

Thanks!

https://mydramalist.com/article/2017/01/20/5-classic-romance-jdramas-that-you-need-to-watch

  • Hana yori dango – it’s a must see it’s the classical Cinderella story but one of the bests ever done (2 seasons and 1 movie)
  • Buzzer Beat – good romance drama about 2 people that meet and fall in love but first they have to find the perfect timing for them to be together 
  • Love Shuffle – it’s a story involving 4 couples and how they learn what love is
  • Pride – a classic romance drama with a mature couple who grow and learn from each other and also heal each others wounds
  • Tatta Hitotsu no koi – a very good romance and it’s a Cinderella story but inverse (rich girl, poor boy)
  • Hotaru no Hikari – this has 2 seasons and a movie, and it’s a bout a couple that has to live together due to some circumstances and gradually fall in love
  • We married as a job - As mentioned before this one is compared to Because this is my first life. I liked both of them but I'll have to say I liked better how they managed the "contract relationship" at the final episodes here. 
  • Soratobu Kouhoushitsu - I like this one, because is also about 2 persons who had their dreams broken about their ideal jobs but with the help of each other discover a new passion in life. 
  • Kahogo no Kahoko - This one is pretty sweet and lighthearted, the OTP is very cute and it centers alot on family.
  • Boku unmei no hito desu - This one is also pretty lighthearted, the male lead is very sweet and kind and it's totally focused on romance

AS for your main topic "Out of their league"

  • Takane no Hana
  • Densha otoko
  • Black prince and wolf girl

Zenkai Girl (technically they're both from working-class background, but she aspires to be a top corporate lawyer, while he seems to be content just working in a small food joint and raising his kid)

  • Closest Love to heaven

Most of the ones I know are Korean 

For out of league romance, i second Takane no Hana and Densha Otoko

Other than that:

Star no Koi - cliche story about common man falls for popular star (actress)

Yamato Nadeshiko - poor man falls for "high maintenance" girl

Celeb to Bimbo Taro - again poor man falls for rich girl

Bus Stop - bus driver falls for career OL (office lady)