You MIGHT be able to consult the archive in a university library. Most university libraries will allow members of the public to use their reference facilities if no comparable sources are available in local public libraries. (I've used the University of Essex library, simply by 'signing in' at the reception desk, on dozens of occasions).
However there can be problems with subscription services. While universities generally allow you to consult any of their books, some subscription services impose a condition that the institution paying for the service will only permit bona fide students and staff from that institution to use the service.
Even so, it might be worth a phone call to your nearest university library. Otherwise you'll have to ask your local public library service whether they've retained the printed copies of The Listener (which is a possibility) or subscribed to the online service (which, given financial cutbacks, is unlikely).
As a last resort (if you're close to London) you could apply for a reader's pass for the relevant part of the British Library. (The main collection is at St Pancras but the newspaper collection is at Colindale. You'd need to ask where copies of The Spectator are held):
http://www.bl.uk/
(You should note that the British Library only issues passes to readers who can show that the materials they're seeking aren't readily accessible elsewhere; so it's important to try the other routes first).
Chris