How can I start an Activity from a non-Activity class?

Once you have obtained the context in your onTap() you can also do:

Intent myIntent = new Intent(mContext, theNewActivity.class);
mContext.startActivity(myIntent);

Your onTap override receives the MapView from which you can obtain the Context:

@Override
public boolean onTap(GeoPoint p, MapView mapView)
{
    // ...

    Intent intent = new Intent();
    intent.setClass(mapView.getContext(), FullscreenView.class);
    startActivity(intent);

    // ...
}

I don't know if this is good practice or not, but casting a Context object to an Activity object compiles fine.

Try this: ((Activity) mContext).startActivity(...)

You can define a context for your application say ExampleContext which will hold the context of your application and then use it to instantiate an activity like this:

var intent = new Intent(Application.ApplicationContext, typeof(Activity2));
intent.AddFlags(ActivityFlags.NewTask);
Application.ApplicationContext.StartActivity(intent);

Please bear in mind that this code is written in C# as I use MonoDroid, but I believe it is very similar to Java. For how to create an ApplicationContext look at this thread

This is how I made my Application Class

    [Application]
    public class Application : Android.App.Application, IApplication
    {
        public Application(IntPtr handle, JniHandleOwnership transfer) : base(handle, transfer)
        {

        }
        public object MyObject { get; set; }
    }