If you are arriving with Ryanair, take the airport coach to Stockholm.
More information on that can be found here: https://www.flygbussarna.se/en/vasteras
The airport coach takes you to the main bus station. Next to the bus station you will find Stockholm’s main train station. From there take the commuter train towards Märsta or Uppsala and get off at Solna, it is two stops from Stockholm City. From Solna commuter train station you can either walk 1.1 km to the Winery Hotel or take the bus. If you elect to walk, exit the train station from the exit that lies in same direction as the front end of your train. If you want to take the bus, exit the train station from the back (again, this is in relation to the direction in which your train will be travelling) entrance and cross the road using the pedestrian underpass. Here you will find a bus stop from where you can take bus 176 or 177 towards Mörby Station. Get off the bus at the stop Järva Krog. The Winery Hotel will be in view across the highway. Use the pedestrian underpass next to the bus stop to cross the road.
More information on public transportation in Stockholm is available here: https://sl.se/en/
There is no rail service from Västerås airport. You can however take a local bus from the airport to the Västerås train station and catch a train from there towards Stockholm. All in all, this will take about 10 minutes longer than the airport coach, but it will also be a little cheaper. Travelling to the Winery Hotel you should get off the train at Sundbyberg Station (a Stockholm suburb) rather than at the Stockholm Central Station. From Sundbyberg take bus 509 towards Danderyds Sjukhus and exit the bus at the stop Rosenborgsgatan.
Due to construction work on the railroad between the 13th of July and the 5th of August rail service will be suspended on part of the Västerås-Stockholm leg during that time and replacement buses will instead be used for part of the journey. The airport coach is thus the only recommended means of transportation between those dates.
Use the Resrobot site to buy tickets for the local bus from Västerås airport and the train onwards to Stockholm. Input “Västerås flygplats” in from and “Sundbybergs Station” in to: https://kopbiljett.resrobot.se/en/#/