You can do it on your own, of course. Public transport is available to bring you to any starting points of your tour.
But you should check in beforehand where the little Heurigen are, and more importantly, when they are open. The smaller establishments have only certain dates (or weeks) during the season for offering wine and food.
If you are tough you can make the full route 1 (Stadtwanderweg = city hiking trail). Better is to go by bus (38A) to the top of Kahlenberg and do just the downward part.
https://www.wien.gv.at/umwelt/wald/freizeit/wandern/wege/wanderweg1.html
Also nice:
https://www.wienzufuss.at/route/emil-und-elise-streckenwanderung-von-sievering-nach-grinzing/
Internet pages are in German only, but when using Google Translate easily understandable.