It's not web-based - it pushes your email to your phone just like you're looking to do. What you do is have your mail forwarded to your mail2web email address (yourname@mail2web.com) and it's then pushed to your phone. What I did with my optonline.net account is log in to my optonline account on the web and set it up to forward a copy of my emails to my mail2web account. The only problem with that is that when you reply from your phone, it shows your mail2web email account as the account you are replying from; however, if you use the paid service (I pay $4.95 per month), you can set it up so that your replies come from whatever email address you want. So basically when someone sends an email to me at
myname@optonline.net, it comes to me on my phone and if I reply from my phone, it comes from
myname@optonline.net. It's actually perfect. Not sure if it will work the same with your setup but you should look into it.
Check out this link for some info:
http://lifehacker.com/398526/set-up-push-email-contacts-and-calendar-on-your-iphone-for-free