From using that train fairly frequently, there is a number of things that would help journey times. The obvious, easy wins would be to run it from Great Victoria Street (which will obviously happen), and make it an express service. That alone would shave 10-15 minutes off the time.
The line through Lurgan and Portadown has some quite tight turns, and the service is very slow through there. Likewise through Newry, and the viaduct before it, it's very slow there too. Down south, it's very slow through Drogheda, again some tight corners and a viaduct. Chopping off these sections from the route would help, such as below.

North1 by
hypnotoad24, on Flickr
South1 by
hypnotoad24, on Flickr
The slowest bit of the service is still the section into Dublin, so additional track would be needed, or a re-prioritisation of the services using the track.
To get a 60 minute service would require an entirely new line from Belfast to Dublin to be built i imagine.