There are two questions here.
1) Azure Web Sites does support .NET 4.5.1, as in the update is installed on all Azure Web Site VMs. This has been available for several months now. So when you select 4.5, you are really running 4.5.1. The version reading "4.5" in the portal is misleading, unfortunately, and due to the fact the 4.5.1 upgrade did not change major/minor versions of framework so the "4.5" remains.
2) The specific "ASP.NET application suspension feature" unfortunately is part of a .NET 4.5.1 version that only ships in Windows 8.1/Server 2012R2. Azure Web Sites VMs currently only run Server 2012. Therefore ASP.NET application suspension is not supported in Azure Web Sites.
Azure Web Sites does other things to attempt to manage cold start performance issues (such as aggressive caching of compiled assemblies) but this is certainly a feature we would like to add in a future version of WAWS. Thanks for the feedback.