"Ar" together with verbal nouns

The preposition ar as used in modern Irish is actually a fusion of several different prepositions. One of those was iar meaning "after", which, as you should know, still exists as a prefix used in word-building. Some usages of ar are obviously based on iar.

In Irish, we usually use tar éis and i ndiaidh (as well as the many dialectal variants of those, such as ar théis, thar éis, thar n-éis and in éis) for saying "having done something". See these examples:

Tar éis dom an obair a chríochnú, chuaigh mé abhaile.
I ndiaidh dom an obair a chríochnú, chuaigh mé abhaile. 

Both mean "having finished the work, I went home". Note that the preposition do is used in these constructions to refer to the agent of the verbal noun construction. Compare:

Tar éis dom an obair a chríochnú, chuaigh mé abhaile.
I ndiaidh dom an obair a chríochnú, chuaigh mé abhaile.

"Having finished the work, I went home."

Nuair a bhí an obair críochnaithe agam, chuaigh mé abhaile.

"When I had finished the work, I went home."

Chríochnaigh mé an obair agus chuaigh mé abhaile.

"I finished the work and went home."

These days colloquial Irish probably prefers Tar éis mé an obair a chríochnú...I ndiaidh mé an obair a chríochnú..., and that is not wrong either.

But now to ar < iar. My impression is that it is falling into disuse, although even very colloquial-based texts do include such constructions as ar theacht abhaile dom = when I had come home, having come home. (Actually, you also do se ar dteacht abhaile dom, with eclipsis. That is a survival from the good old days when ar and iar were still distinct prepositions, because iar used to eclipse the following noun.)

The ar 'after' constructions with verbal nouns are crucially different from the tar éis and i ndiaidh constructions, because the object comes after the verbal noun, in genitive form:

ar chríochnú na hoibre dom.

Compare:

ar chríochnú na hoibre dom
i ndiaidh dom an obair a chríochnú
tar éis dom an obair a chríochnú
i ndiaidh mé an obair a chríochnú
tar éis mé an obair a chríochnú.