I am afraid it is going to be hard to find local tour operators, there is not really a market for it. It is possible to visit Egeskov on your own with public transport, but is more than two hours each way. www.rejseplanen.dk is the travel planner to use for all public transport in Denmark, type in Fredericia st. and Egeskov Slotspark as your departure and destination for correct results.
If you are not keen on spending 300$ or two hours onborad trains and busses, there are other interesting sights in the vicinity of Fredericia. Koldninghus castle http://en.wikipedia.org/wiki/Koldinghus in nearby Kolding is a proper castle, not a manor house/amusementpark like Egeskov. It is less pretty but has more history and Kolding as only 15 minutes away by train.
Another option could be the Jelling Rune Stones http://en.wikipedia.org/wiki/Jelling_stones in Jelling, made around 935 a.d. the rune stones are considered the birthcertificate of the Kingdom of Denmark. Though they are not very grand or impressive ,they are a unesco world heritage sight and hold great historic significance.