Update: I originally said these characters were obliterated from the language. I was wrong about that, as shown in the accompanying picture.
As you can see in the picture above, the character does pop up now and again. I've probably passed by this place in Shinjuku countless times, but never took notice of it until recently when this question had the issue of archaic characters on my mind. So I took a picture with the intent of seeing how people reacted to it.
First, interestingly, it seems from looking up the restaurant on the web (look at the grey furigana above the restaurant name on the linked site), they intend for the middle character,
ヰ, to be pronounced as
イ, to form
ワイン, as in "wine".
So the restaurant is called
新宿ワヰン酒場 and read
しんじゅく ワイン さかば. "Shinjuku Wine Bar".
However, this is obviously done for very deliberate effect, and only by the context of obviously being a wine bar does the reading deviate from its original pronunciation and take on a modern one.
So it is not read
ウィ, as I would have assumed. When I first saw it, I thought it would be read
ワウィン, which is a torturous word to pronounce. I think without context like this restaurant, the characters would usually be read
ウェ, respectively, but their official removal from the language means they are subject to individual play.
I asked some Japanese people around me (some of them strangers in the Starbucks I happen to be writing this at) how they would tell a friend over the phone about this restaurant so that the listener would know the middle character was not
Some people at first said that it was simply not possible to describe it over the phone, that it had to be shown. One suggested it might be similar enough to a Korean character to use that as a way to explain it, which is obviously not really a viable method but gives a sense of how unfamiliar these characters have become.
However, after a little while and some group consensus, one description emerged as a clear winner:
Even the people who at first didn't know how they would describe it said that if someone said that to them, they would know what was being described, even if they might have to look it up to confirm its exact shape.
Thus, I think if you really wanted to be understood among people who weren't オタク about writing and characters, saying
「昔の...」 would be the description of choice.