Bazantine

Pandora Hearts has a character sort of like that, but the manga is not focused on them.

http://www.mangago.zone/read-manga/three_days_of_happiness/
I think this one fits pretty well! The MC is a loser in life, sells a fat chunk of his lifespan with only a few months left. Uh, hardly anyone knows besides the people involved in the transaction, since people would think he's a loon if he goes on about selling some of his years. He doesn't really know what to do in the beginning but he steadily changes and lives happily and carefreely in his last days ヾ(❀╹◡╹)ﾉ~

Toki doki (one shot) ٩(๑❛ᴗ❛๑)۶

http://www.mangago.zone/read-manga/toki_doki/ is a really good oneshot that fits !

http://www.mangago.zone/read-manga/10_years_where_i_loved_you_the_most/

http://www.mangago.zone/read-manga/kowagaranai_de_soba_ni_ite/

http://www.mangago.zone/read-manga/kashikomarimashita_destiny/

December Rain
Epitaph (Perfume)

http://www.mangago.zone/read-manga/kimi_wo_watareba/

୨୧ UNREQUITED, TRAGEDY STORIES ୨୧
http://www.mangago.zone/read-manga/bokura_wa_nando_mo_koi_o_suru/
http://www.mangago.zone/read-manga/tamayura_yuki_ringo/
http://www.mangago.zone/read-manga/hana_no_miyako_de/
http://www.mangago.zone/read-manga/tadayoedo_shizumazu_saredo_naki_mo_sezu/
http://www.mangago.zone/read-manga/yume_yume_shinjuu/
http://www.mangago.zone/read-manga/double_mints/
http://www.mangago.zone/read-manga/sakura_gari/
http://www.mangago.zone/read-manga/hana_no_migoro_ni/
http://www.mangago.zone/read-manga/kanashii_kotoba/
http://www.mangago.zone/read-manga/matsugo_no_yume/ (oneshot)
http://www.mangago.zone/read-manga/senya_ichiya_shitone_no_himegoto/
http://www.mangago.zone/read-manga/ouji_to_kotori/

if you do like these recommendations, feel free to follow some of my other lists!
http://www.mangago.zone/home/mangalist/615664/
http://www.mangago.zone/home/people/728364/home/

i hope you have a good day today! (๑･◡･๑)