With a flight out of Milan at 1pm, staying in Varenna the night before is doable if you want. It's a one hour local train from Varenna to Milano Centrale, then you could take the Malpensa Express train right from the central station to get to the airport. Allow another 50 minutes for that connection (there are two lines: Malpensa-Milano Cadorna is a quicker 35 minute trip and Malpensa-Milano Centrale takes about 50 minutes). You will probably want to plan on arriving at the airport between 2-3 hours prior to your flight. Personally, I would stay in Milan the night before the flight just to lower my risk for potential connection issues, like train strikes or delays. On the day I fly out I always try to play it safe and choose the most direct route to make for an easy travel day.
As far as taking the train or shuttle to the airport, Rick opts for the train over a shuttle bus, but either will do. Here's what he explains in the Italy guidebook:
The Malpensa Express train is usually the most sensible way to travel (not covered by rail passes, tel. 800-500-005, www.malpensaexpress.it).
I prefer the train, which is speedy, reliable, and reasonably priced. But two bus companies run between Malpensa Airport and Milano Centrale train station, offering virtually identical, competing services. They each charge about €10 for the one-hour trip (buy ticket from driver) and depart from the same places: in front of Terminal 2 (outside exit 4) and from Piazza Luigi di Savoia (on the east side of Milan’s central train station--with your back to the tracks, exit to the left). They also pick up and drop off at Terminal 1, which, if your flight docks there, might be a reason to prefer the bus. Buses leave about every 20 minutes, every day, from very early until just after midnight (Malpensa Shuttle tel. 02-5858-3185, www.malpensashuttle.it; Autostradale tel. 02-3391-0794, www.autostradale.it). They’re almost comically competitive, regularly offering deals such as three rides for the price of two.
Check out Rick's Italy book for hotel recommendations. Hotel Florida (www.hotelfloridamilan.com) and Hotel Garda (www.hotelgardamilan.com) are two we like near the train station. Buon Viaggio!