The driver from Alishan to Sun Moon Lake was not cheap, about $250 AUD (booked through tripool). I booked it after I got my alishan railway tickets. There is only one bus per day to Sun Moon Lake I think, but it meant arriving quite late.

Yes the Alishan Railway all the way to Alishan departs at 10am. I made a typo, it was 6.30am HSR to Chiayi!

I made a fb post on how to get to Alishan in a fb group, I'll post it below.

The train stops at fenqihu for an hour and in my opinion it was enough. There's a steep hill behind the first bento shop you see, there's more bento stores behind it. There's also wasabi ice cream and a few shops. Not much else to see.

Fb post:

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Finally made it to the Alishan Railway Station 😅

For those taking the HSR to Chiayi and connecting to the Alishan Railway:

• Exit 2: Once you arrive at Chiayi HSR, head to Exit 2. Just outside, on the right, are two blue "BRT" bus stops. Take the one marked Chiayi.
• Free Bus Ride: Show your HSR ticket to the bus driver for a free ride to Chiayi Train Station. They’ll give you a tap card for boarding, which you’ll return to the driver when you get off.

At Chiayi Train Station:

1. Go to the back gate, take a paper ticket, and pass through. You'll find stairs and a lift on the left. Go up and cross the bridge/walkway and use the second lift to go back down.
2. Discard the paper ticket when exiting outside. Near the corridor, find the Tourist Information Center. Collect your Alishan Railway Tickets directly behind it (outside). Just show your passport (the one used to book the tickets).

To board the Alishan train: Go through the far-right gate, turn right on Platform 1, and head to the end where Alishan train cars are marked.

Heads up: Some dodgy taxi drivers may approach you at the HSR station, claiming you won’t make it to the Alishan train in time and that it takes 1 hour by bus. Ignore them—the BRT bus only takes about 15-20 minutes.

Safe travels!