It is my understanding that a web administrator is the person who ensures that the space that clients use for their websites is available, operational and highly functional.
In general, an administrator can and does wear many hats. Sometimes they have to do a little IT (information technology) work, for instance if the server that the website sits on or if the cloud environment that contains the website set-up is having problems or is down. The administrator may also wear a designer hat meaning they may have some responsibilities on either designing from scratch or building a website design or changing a website design based on the client's needs. The administrator may also be in charge of costs for maintaining a website. This can be a biggie because whether the client has their website on a server or in the cloud, there are costs associated with usage, uptime, number of users, additional technical support when needed, or any number of items that are used to determine how much money it takes to keep the website up and functional.
As you might imagine from the list I have shared, wearing all of these hats can lead to a large number of important experiences that will be valuable to a person that takes on this sort of role. Cost management, client management, being a great listener, ensuring that the site is up and functional for a long period of time (lots of website crashes is a sure fire way to lose customers), adding to your technical support skills...these are all great experience that I imagine one could get if they decide to take on the role as a web administrator.
Hope you find this answer helpful and best of luck to you!
"what kind of experience can you attain while working as a web administrator?"
Well it all depends on the scope of the administration part. There will be some web administrator that simply keeps the site healthy, dealing with providers, making sure the server on which the website is works and is secured and so on.
There are web administrators that also keep the web site's content up-to-date. Finally some will even code web application, to improve end-user experience using web technology language.