Sorry, the communication got muddled.
While you are in Gyor you might want to ask the hotel for transportation to the nearby town of Pannonhalma where you will find a most beautiful and ancient Archabbey. https://bences.hu/lang/en/ Worth the effort.
THEN, when you get to Budapest, if you want some help on getting to your AirBnb from Keleti; it would be a pleasure.
If you are indeed staying near Nyugati, then its half way across town from Keleti. If you have a lot of luggage, I would just call a taxi. I like City Taxi http://www.citytaxi.hu/ They have always been honest, prompt, and pleasant. When you call you will hear a recording in Hungarian, then a recording in English, then someone will come on the line in Hungarian. At that point you say "Good Afternoon" or what ever the time dictates, and they will switch to perfect English.
If you do want to go with public transportation, your options are two:
From Keleti take Metro Line 2 one stop to Blaha Lujza ter. Get off and get on the 4/6 Tram for about 4 stops to Nyugati. Very scenic way to get there. However this tram moves more passengers than any other tram line in Europe and it can be a bid crowded. Option 2, take Metro Line 2 three stops to Deak Ferenc ter and then change to Metro Line 3 and its two stops to Nyugati. Deak Fernec ter is the central hub for the three main metro lines and it can be a bit busy but its an adventure. The Nyugati metro station is sort of to the back of the station and you will have to either walk through the underground (very "interesting"), walk through the station, or walk the street quite a ways to the front of the station.
You could also take the new Metro 4 line a few stops to where it crosses the M3 and then on to Nyugati.
Take a Taxi