The use of "came" baffles me here.

There was, of course, a vague uneasiness about it all, but I would able to handle that when i ‘came’ to it. is it good or correct to use “came” as being used here.And does it means he has dealt with the problem or he is now going to deal with.please assist

Yes, “came” is correct.

“when I came to it” = when I encountered that situation (in the future)

Try to remember three things:

Put a space after full stops and commas.
Start each sentence with a capital letter.
Always capitalise the word “I”.

These things will make a BIG difference to the quality of your writing, for very little effort.

But Dozy, I thought came talks about things in the past why it’s alright being used here rather than ‘come’. Please confirm. Thanks

“I will be able to handle that when I come to it” describes the speaker’s predictions now about events in the future. The past-tense version of this statement is “I would be able to handle that when I came to it”, which describes the speaker’s predictions about events in the future then, but in the past now. Since the narrative you quote from is describing past events in the past tense, the “would … came” version is appropriate.