Coming up,gone up Vs Coming ,gone

If there is a different elevation, (as in a hilly location) you can use "up" and "down."
Sometimes "down" is toward the city center, as in downtown and uptown.
Sometimes "up" is approaching, and "down" is receding.

She is coming up the hill.
He is running down the street.
Ram lives down the block from us.

2.He has gone up to buy your outfit. This is not

All the sentences are from Oscar Wilde's book The importance of being earnest which I read a few days ago.

In that case, 'He has gone up to buy your outfit' could mean that he has gone to London. People used to go 'up' (to London) and come 'down' (from London)..

Can we use such sentences in day-to-day language.

Yes. AS explained how they could have meaning.

fatimah0786All the sentences are from Oscar Wilde's book

If you had mentioned this in your post, we could have given you a better answer.
Oscar Wilde writes in a style of a Londoner's late Victorian British English. The society and language have changed since then.

Why do we say 'down the street' rather than 'on the street'.

We use 'up' or 'down' when we are talking of directions.

Thanks to all for answering . The thing that up and 'down' mean directions in sentences like,Ram lives down the block from us

fatimah0786Ram lives down the block from us

He lives about one block away. That is a short distance.

fatimah0786He is running down the street

Actually, "up" and "down" are the only prepositions that fit to indicate the direction he is going in.
Depending on the context, down could be:

down = away from you
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