The answer-- it depends.
The first step in looking for an apartment is to have pre-approval. This is because in this hot market, if you see a place you like, you simply don't want to waste time getting one after, because by then it may be gone. Second, as many have said, no use wasting agent's time or yours if you don't have it. If it's an open house, go ahead and window shop. Yet, on something like a huge price tag, many sellers won't even consider having the door opened unless you're a recognized name or are pre-approved or cash. For example, on a $49 million listing in Tribeca, the first question I had to ask the customer was for this, because you can't go window shopping these kinds of listings without.
Getting a pre-approval means you're serious. And, yes, you can use the same one for different apartments. However, a rookie mistake I see is not having or making it adjustable depending on your offer. In other words, looking at a $1 million apartment and offering $900,000? The letter shouldn't say you're preapproved for the million, because the seller will want that million. Makes it hard to negotiate.
Need help? I'm here for you, and a local resident and agent in Brooklyn Heights, my family's chosen neighborhood. AND I rebate 10% of my commission to customer within 2 wks. of closing. On a $1 million property that's approximately a $2500 rebate.