I'm trying to find a Japanese book titled something like "When all the cats in the world had died" (translation may be inaccurate). I tried Googling that English title-- no luck.
The story is about a man awaiting his death. The devil offers him one extra day to live, for each important thing he gives up doing everyday.
The book was most probably published in 2013, is a short story and the cover image is grey colored with a cat picture on it.
With judicious use of Google Translate and Amazon Japan, I believe I've found your book.
世界から猫が消えたなら "If cats disappeared from the world"
川村 元気: Kawamura Genki
More info on Amazon. In case the link doesn't work, copying and pasting it into Amazon Japan's search box should work:
https://www.amazon.co.jp/川村-元気/e/B00A7LUCD2/ref=ntt_athr_dp_pel_1
The cover fits (cat peering over a wall, gray-toned) and it was published in late 2012:
If A Cat Disappears From The World also known as Sekai kara Neko ga Kieta nara is a live-action movie that will be shown in Japan in 2016 starring Takeru Satô.
Takeru Satô is a Japanese actor best known for playing the role of Himura Kenshin in the Rurouni Kenshin live-adaptation film and its sequels.
From IMDb:
Young mailman finds out he has no time left due to a terminal disease. Suddenly he is approached by a devil that offers him to live more time if he eliminates something from the world. Young mailman then thinks about his relationship with friends, ex-partners, family.
