You can get there by also by bus (check Eurolines) or train, but the best options are the ones you already mentioned.
I wouldn't recommend taking a bus. It would be a very long and tiring bus ride all the way from Greece. You would have to change buses since there is no direct bus line. And the price is unlikely to be much better than flying.
Taking the train is a possibility but also very time consuming, you would probably have to change trains multiple times. It is likely to be much more expensive than flying.
I would recommend to you, if you are unable to get any reasonable flights to the nearby airports, to opt to fly to Copenhagen and take the train from there to Horsens. Anyway, you are more likely to find a good deal and a direct flight to Copenhagen (from Athens at least). The train ride is smooth, takes typically less than 3 hours, normal price without seat reservation: 362 DKR. The trains are fairly frequent, usually one per hour, except during night. Check DSB or the Journey Planner (in English) for timetables/prices and further info.
If you have a car you could of course also drive there yourself, but it's hardly worth it for one person (fuel costs, extra time spent driving). Only maybe if you were planning to do a lot of travelling around once you are there.