I don't know where they got that date from, but it's incorrect. Dumbledore was born in 1881.
Albus Dumbledore - Wikipedia, the free encyclopedia HPL: Albus Dumbledore J.K.Rowling Official Site
Making him 114-115 when he died. Probably 114 turning 115 right before he died:
"Birthdate: 1881 (JKR), probably in July or August. This date supersedes Rowling's statement in 2001 that Dumbledore was "about 150 years old" (Blue Nose Day) and fits better with the dates that appear in Book 7. Regarding his month of birth, Rita Skeeter states in The Life and Lies of Albus Dumbledore that Dumbledore was "nearing" his 18th birthday when he left Hogwarts in June, but was still 17 when he met Grindelwald (DH18). Basically, at some point during Grindelwald’s stay at Godric’s Hollow Dumbledore turned 18, so the month of his birth appears to be July or August."
The 1844 date is probably based on Rowling saying he was about 150 in 2001.
The Ron quote was just a joke.