I am not sure what difference there is between
- This is the first time I've lost my passport.
- This is a first time I've lost my passport.
Would appreciate your explanation.
This is the first time I've lost my passport.
The first time I went to London, I lost my passport.
The first time is a specific time, hence the definite article. “*
A first time” would imply that there are many first times.
It is also possible to say my first time, although that is mostly done when there is no verb.
This is my first time losing my passport. [Possible but not very common]
This is the first time I've been to London.
This is my first time in London.
A first time is rare. It can be used in when referring to the concept of doing something for the first time, with no specific thing in mind.
There's a first time for everything.
When referring to a specific circumstance that is encountered for the first time, you can say a first (for me).
Losing my passport was a first for me.
First-time can also function as an adjective, to mean a person who is doing something for the first time: a first-time voter (someone who is voting or will vote for the first time in his life), a first-time writer (someone who is writing his first book), …
"This is a first" is a standalone phrase, employed to convey that whatever is happening constitutes an experience that is completely novel to the speaker in some way.
The phrase uses "first" as a noun, specifically in sense 3.2.a. in The Merriam-Webster Dictionary:
something that is first: as....the first occurrence or item of a kind
In the story, Kafka has the thought "this is a first" in reference to having to "pick up the scattered jigsaw puzzle pieces of [himself] lying all around." He immediately reconsiders, however, and remarks that he "had this feeling somewhere before". Without having read the whole book, I can't say for certain what other experience he might be alluding to, if any was even specifically mentioned earlier on.
What it all amounts to is that Kafka has experienced something in his past that feels similar enough to what he's doing now that he's not sure he should be saying "this is a first", even though he's doing something that appears initially to be completely new and strange.
"This/that is a first", colloquially, is an idiom that means "this is the first time this has ever happened (to me)." It gives force to the idea of experiencing something for the first time, in a short, succinct manner. Essentially a first means "the first time/experience/occurrence."
If it were "This is the first," it actually wouldn't make sense, because that phrase begs the question: "The first what?"