He specificly asked about hosting the app on Linux, so let us assume that Windows Server licensing is not an option here.
I used Mono in the latest version to run .NET 4.5 apps, it is far more along than your comment would lead one to believe. Always check the source, the distribution repos are quite behind on both Wine and Mono, both are better than you know.
FWIW: I ran some google searches, and found some GREAT how-to information. The best information about running under Wine with .NET 4.5 was on the WINE site itself, the rest are all a bit dated or incomplete. Nonethless, I would read several and then follow the official one with more tricks and knowledgte to fall back on.
NOTE1: Do NOT try to mix Mono and .NET installs, they so not always play welll together! Try Wine with .NET, or Mono on Linux without Wine: not both at the same time.
is the best place to start with Wine. Mono is at www.mono-project.com
and is today in release 3.12.1. Official and up-to-date packages are available for
Debian, Ubuntu, and derivatives
CentOS, Fedora, and derivatives
openSUSE and SLES
(Also Windows and Mac)
Third party and maintainer packages are available for all of those and many more, but those are rarely current versions.
It not only looks doable to host your application on Linux using one of these solutions, but it looks pretty easy! I plan to retest my (successful) earlier applciation testing using the new tricks and see if it works better, faster, and easier than when I started.