I will be lucky enough to be in Stockholm on Midsummer's eve this year, and would be willing to travel to nearby areas.
Are there public Midsummer's Eve events near Stockholm I could attend?
Yes there are plenty. The most thourough experience is probably to visit Skansen - the outdoors combined zoo and open-air museum at Djurgården. They have some shorter information in english about this. In swedish there are several pages about the festivities.
Almost everwhere there are some kind of local festivity going on. It may be hard to track down. In Sollentuna where I live there is traditional celebrations at the local folk museum (called hembygdsgård in swedish). This is very common.
Most swedes however bring friends and family to their summer houses and have a smaller celebration. This is the most genuine way to do it. Things to expect:
Then you may either have a maypole of your own to construct and raise or you visit one of the local festivities for this part of the celebrations. Once the pole is raised there are supposed to be some music and tradiational dancing around it. This part is mostly forgotten and usually only includes the childrens dance called "Små grodorna" - the little frogs. (Skansen will surely have some traditional dancing going on though.) Update: a video clip to get the feeling for this part.
After the pole part it may be time for sports. Kubb is a game from gotland that is played all over the country nowadays. I have also been involved in soccer games at this stage of the day.
If in doubt about any of this just ask any swede at hand. They will be happy to explain it all.
In short - a bigger celebration may have more of the traditional stuff included but a smaller one is more the way most swedes do it.
After this I leave you with this (somewhat sad) maypole from my families celebrations in 2011.