<<"Will it be fine if I were to wear my normal running shoes and walk around the region?">>
Yes and no. ;-)
Depends on what you mean by "walk around the region."
There are trails and walks of varying difficulty.
On this link, you will note that some trails show a red and white indicator, while others are yellow.
www.jungfrau.ch/en/som mer/tourism/destinatio ns/kleine-scheidegg/hiking-trails/
This link explains the types of trails:
If the trail has a red/white marker, you need boots or heavy sturdy shoes with lugged soles and a hiking staff.
Yellow-marked trails are OK with running shoes, but even the "yellow" trails can be rocky and rough, so that thin soles may let your feet feel more of the surface texture than you like.
The link gives some good perspective and explanation, especially for the more difficult hikes:
Here are some pictures (attached) of different portions of an easy hike. There are level and also somewhat downhill portions, even on an easy (yellow coded) hike. It is near the end of the Panoramic Trail from Männlichen to Kleine Scheidegg. That is sometimes called "The Grannies' Walk" because it is so easy. ;-)
Notice the gravel underfoot. If you try to hurry on that kind of surface, it is easy to slip. If you take your time, no problem.
The last picture, showing the short climb to Männlichen Gipfel, is a "red/white" "mountain trail."
Also, those easy trails can often be wet, where snow melt runs across them. Running shoes are not waterproof....