Ajax is very viable for web development. It is built into frameworks like jQuery. It may be confusing though because you see the term used in things like the asp.net ajax control toolkit, which do a lot of interesting things client-side. A lot of those client-side things like the modal popup exist in other scripting libraries like jQuery UI and Bootstrap.
Separate the real purpose of AJAX which is to perform asynchronous communication with the server and ignore the other items that get confused with AJAX (like the aforementioned AJAX control toolkit).
Bootstrap is a responsive design framework. It has lots of built-in components such as the modal, tab, alert, etc.. Don't confuse these with AJAX either, they're just great client-side functions. The AJAX control toolkit also has similar features, but few parts have much to do with true AJAX other than the fact they make use of scripting features introduced in 2008 with the System.Web.Extensions namespace. Things like the $find() method.
Look at Bootstrap to focus on the UI, it's presentation, and some interaction components. Use the parts that make sense to you. If you have an Ajax Control Toolkit modal that works in your application just fine, you don't need to replace it with the Bootstrap modal as you may need to do other things to get the two to behave the same. Bootstrap is an awesome framework for client-side development, and can easily work with ASP.Net, though it's easier to do with MVC simply because of the design principles of MVC versus Webforms.