I've heard this in an anime but need to understand the definition of it
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Since there isn't enough context to judge exactly in what sense it is used, I guess it was similar to この私が〜します. And I also assume that you wonder why there is この before 私.
Here, "Kono watashi ga (この私が)" basically means "I", but it is emphasized. It is also pronounced as "Kono watakushi ga" in more formal way.
If I was 20 years younger, I'd marry her myself. (Here, not others but I get married to her)
You are surprised at my wishing to postpone it.
If it is OK, shall I go get it?