Place A in the standing position
Is it correct to say place A in the standing position? Is it also correct to sa in a standing position?

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You will get a good answer only with the context

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Place the statue  in A standing position/ upright position in the storeroom , please .

In the standing position > This is not often used unless the object has many ways it can stand.
> Place A in the standing position that I showed you just now.

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In a description of a procedure.

For example, 
First, place the machine in the standing position.
Second, detach the part A.

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Can you provide further context?