The man in this rhyme is sometimes called Old Abram Brown and sometimes he's called Old Grimes. This is a nursery rhyme and also a round that's sung by choirs.

Old Abram Brown is Dead and Gone - English Children's Songs - England - Mama Lisa's World: Children's Songs and Rhymes from Around the World  - Intro Image

Notes

Interestingly, I found the following in Familiar Quotations (1875) by John Bartlett:

John Lee is dead, that good old man,-
We ne'er shall see him more:
He used to wear an old drab coat
All buttoned down before.

An inscription in Matherne Church-yard, 'To the memory of John Lee, who died May 21, 1823, aged 103 years".

It seems that the graveyard inscription was quoting the rhyme/song. 

According to The Oxford Dictionary of Nursery Rhymes, "Old Abram Brown" is from a mummers' play.  The mummers would go around from house to house in disguise and perform plays.  Mumming goes back to the Middle Ages and was extremely popular in Great Britain.  Mumming was often done at specific times of the year, like Easter, Christmas and All Souls Day (near Halloween).  Mummers' plays often involved the killing and resuscitation of one of the characters.

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Here's another version of this rhyme from The Real Mother Goose (1916), illustrated by Blanche Fisher Wright.

Old Grimes is Dead, That Good Old Man

Old Grimes is dead, that good old man,
We ne'er shall see him more;
He used to wear a long brown coat
All buttoned down before.

Old Abram Brown is Dead and Gone - English Children's Songs - England - Mama Lisa's World: Children's Songs and Rhymes from Around the World  - Comment After Song Image
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Sheet Music

Sheet Music - Old Abram Brown is Dead and Gone

Thanks and Acknowledgements

This rhyme and the 1st illustration can be found in The Nursery Rhyme Book, edited by Andrew Lang and illustrated by L. Leslie Brooke (1897). The 2nd illustration is from The Real Mother Goose (1916), illustrated by Blanche Fisher Wright.