Hello,
I am going to try to address most of your questions as best I can.
The word podcast can be confusing, so instead think of it as an audio or video file that is available on the web that people can download and listen to (or watch) when it is most convenient for them. That's it.
What makes a podcast different from just another audio or video file on the web is the RSS feed (RSS is sometimes referred to as Really Simply Syndication). That's the true magic of podcasting. Without an RSS feed, a podcast is NOT a podcast.
The RSS feed allows someone to subscribe to the podcast and have access to the new episodes as soon as they are made available. RSS feeds can be submitted to multiple podcast directories making a show (podcast) available for subscription to hundreds, if not, millions of people.
So, to answer your question: Do I need an RSS feed? The answer is a yes, absolutely.
In terms of actually getting the podcast on the website:
In addition to making the podcast available through an RSS feed on the site, you will also want to post the audio directly onto the website.
For posting the audio file, you will need to ask your client what platform he is using for his site. Meaning, is it Wordpress, Joomla, HTML based or something else? The type of website platform/coding he uses will make a difference in how you post audio online. With Wordpress there are plugins available, etc.
The most simplistic way to get the audio up is to upload the audio file to the web (using a podcast hosting service for the audio is a great way to go and you already know that. I use Libsyn for my shows). Then just create a hyperlink that says something like: Click Here to Listen and then link it to the audio file.
I would say if this sounds like something you are not remotely interested in doing, pass it on to someone else that can help (you can subcontract it out if you want, as well). You want to be happy and confident in the work you provide. If podcast support is not it, that's ok. It took me about 2 years to truly get a solid foundation in all things podcasting and I am still learning.
Of course, if this sounds like something you definitely think you would enjoy, there are plenty of resources on the web and books to start learning. I love podcasting, it's fun. But, it's not for everyone.
If you have any questions, do not hesitate to ask.