I first visited the achilleion in 96 & was last there in july 98. I hope that most of the information I gained then is still relevant.
Local buses do run to gastouri from corfu town though these are few & far between. The best way to get there is definitely by rented vehicle, thus avoiding the dreaded coach party tours. If your budget does not stretch to a car then hire a motorscooter, but do watch the roads as some are like glass in the summertime. The achilleion is accessible from all parts of the island being at a mid-way point. Park on the main road just outside the house, its chaotic but theres usually enough room for all.
The ground floor rooms only were open on my last visit. I thought the building was in good state of repair, the only drawback I could find was there seemed to be a certain lack of exhibits inside. The gardens were excellent & well looked after, and can anyone else lay claim to a better patio area.
On leaving go down the hill heading fo the coast road, this takes you past the Sissi memorial. Then to round off your journey see the strange sight that is kaiser bridge.
The achilleion may not be to everyones taste, some may say its over the top & kitsch, but I think its great. I came away very much smitten, partially with the house, though I seem to recall at the time it was mostly with Empress Sissi.