First of all - welcome to BC !
Wi-fi connection from a computer of any sort to the internet is in two parts. The first is from your computer to the router, the second is from the router to the ISP, usually by telephone line.
The four bars and 72mbps relates to your connection to the router, which appears to be solid. The 'no internet access' relates to your connection from the router to the ISP. It may be you need to log-on to their system in some way.
I work as a volunteer instructor with a local group, meeting twice a week - once in a bar (!) and once in a cafe, using public wi-fi in both. It turns out to be exactly the same system in both places and everybody using the wi-fi connection has to log in with their e-mail address and a password to use it. Have you tried asking the manager how you get on-line?
Of course, if you were able to connect previously but can't now, it may be because their connection is down. Oh, and when you do get on-line there, since it is a public system - don't do anything sensitive while you are connected to it !