How to pass Owin context to a Repo being injected into Api controller

Lets put aside why you have this design and concentrate to the problem: injecting the IOwinContext:

you can also get it from a HttpRequestMessage instance with the GetOwinContext method, however you also need to get a HttpRequestMessage somehow.

Unity does not support injection of the HttpRequestMessage out of the box but you can use a custom DelegatingHandler which stores the current HttpRequestMessage in the container as described here: Inject WebAPI UrlHelper into service using Autofac

The linked question is about Autofac but you can transfer it for work with Unity:

The CurrentRequest and the CurrentRequestHandler can be used from Andrew Davey's answer as it is:

public class CurrentRequest
{
    public HttpRequestMessage Value { get; set; }
}

public class CurrentRequestHandler : DelegatingHandler
{
    protected async override System.Threading.Tasks.Task<HttpResponseMessage> SendAsync(HttpRequestMessage request, System.Threading.CancellationToken cancellationToken)
    {
        var scope = request.GetDependencyScope();
        var currentRequest = (CurrentRequest)scope.GetService(typeof(CurrentRequest));
        currentRequest.Value = request;
        return await base.SendAsync(request, cancellationToken);
    }
}

Then you just need to register the DelegatingHandler with:

httpConfiguration.MessageHandlers.Insert(0, new CurrentRequestHandler());

And register the CurrentRequest and IOwinContext in the container

container.RegisterType<CurrentRequest>(
            new HierarchicalLifetimeManager());

container.RegisterType<IOwinContext>(
    new HierarchicalLifetimeManager(),
    new InjectionFactory(c => c.Resolve<CurrentRequest>().Value.GetOwinContext()));

httpConfiguration.DependencyResolver = new UnityHierarchicalDependencyResolver(container);

Beside the custom delegation handler there are other places to hook into Web.API to capture the HttpRequestMessage for example you can create your own IHttpControllerActivator and use the ExecuteAsync method as described here: Dependency Injection in ASP.NET Web API 2

In a selfhosted application you do not have a HttpContext. You need an other way to move the state around. An option is to use a self implemented HttpContext like:

https://github.com/danielcrenna/graveyard/tree/master/httpcontext-shim