Best way to get to Eilean Donan Castle from Edinburgh

The best way is to hire a car - it's going to be a very long train and bus journey. If you did the latter I would suspect Edinburgh Gateway to Perth, Perth to Inverness, Inverness to Kyle of Lochalsh then bus the last bit. It would probably take 7 to 8 hours even with decent connections.

You'd find somewhere in Kyle of Lochalsh, where you'd get off the train. But there will be various places all along the A87 and in the very nearby Dornie

There are regular direct trains from Edinburgh Waverley to Inverness, for the Kyle of Lochalsh (Not from Edinburgh Gateway), no change in Perth needed.
For the Castle all Glasgow to Skye and Inverness to Skye coaches stop at Dornie, which is very close by.
But yes you would need to stay at the Kyle of Lochalsh, or backtrack to Fort William if on transit.
If driving look at Google maps - many other possibilities.
But really 1.5 days is pushing it, driving or on transit.

@isn31c

There are indeed regular trains from Waverley - and they go through Edinburgh Gateway and Perth. There is little point the OP going all the way from the airport to Waverley when the Edinburgh Gateway train and tram stop is much closer to the airport and they can pick up the same train. It would waste time that can be ill afforded on such a long trip.

If it was me I would hire a small van in Edinburgh (we have used Spaceships) then do the 4 to 5 hour drive out to that area. Camp at the beautifully located Glen Shiel Campground at Shiel Bridge. Head out first thing next morning for the short drive to the castle. Get in early before tour buses and the crowds.
I have seen posts suggesting it isn't worth visiting. Personally I loved my visit. It was getting a bit crowded by mid morning.
If sleeping in a van isn't something you want to do then there is an affordable hostel, Ratagan Hostel not far around the Loch near Shiel Bridge.
If you are one for posh tourism you will find plenty of more upmarket hotel type stuff in Dornie right near the castle.

Edinburgh to Inverness trains never stop at Edinburgh Gateway. The handful of trains from Gateway to Perth go the long way round calling at many stations in Fife. While you are doing that you can be far more productively staying on the tram to Haymarket or Waverley and getting on a direct train on the faster Stirling route.
If it did work as faster to go from Edinburgh Park, then it would be a totally different route, changing into the Inverness train at Stirling, not Perth. The train concerned from Gateway terminates at Stirling.
That is what a timetable check shows.
Given the very limited time Tintaldra's solution sounds the best, then you have your own bed, and can come and go at will.

If you can get to Inverness, these two one-day tours pick up and drop off at Inverness.
Then there would be no need to rent a vehicle.

https://www.rabbies.com/en/scotland-tours/from-inverness/1-3-day-tours/skye-eilean-donan-castle-day-tour

https://www.rabbies.com/en/scotland-tours/from-inverness/1-3-day-tours/torridon-applecross-eilean-donan-castle-day-tour

As I initially sated, the best way is to hire a car.

If you were going to Stirling or Perth then Gateway makes a degree of sense (changing at Stirling for Perth), possibly also Dundee and Aberdeen although they usually reckon the 747 bus is the faster connection for Dundee and Aberdeen via Inverkeithing. But into the Highland main line it doesn't usually work well.
If the OP is arriving transatlantic, a 5 hour plus drive may not be an optimal idea.
Depending where the OP is coming from I was actually wondering if the sleeper from London may work one way, as presumably he is not flying in from the US for 1.5 days. There must be more to this schedule.

We have edited a number of posts. You can provide differing opinions while still keeping it positive.

I stayed in Kyle of Lochalsh for a few days and took a taxi to Eilean Donan Castle, then arranged for the driver to pick me up at a certain time. I took the train to and from Kyle from Inverness. It's a wonderful train journey.

One other solution. That lunchtime train from Edinburgh, if it works on your timing, gets into Inverness at 1706.
There is a coach direct to Dornie from Inverness Bus Station (next to the Railway Station) at 1745, arrives Dornie at 1938.
Then next day you could see the Castle and leave Dornie for Inverness at either 1154 or 1419.
The 1419 gives lots of time at the Castle.
The second one is back in Inverness at 1615, then catch the 1726 train back, change at Perth or the 1853 direct. The latter is back in Edinburgh at around 2230