Can I install an upright handlebar (curved towards me) on a touring bike?
is this a good idea? Or is there some danger that I'm not aware of?


You can safely install different shapes of handlebars on all bikes (when done properly).  If you want to keep your current stem, make sure you buy a handlebar with the exact same diameter as the old one (this is exact to a fraction of mm).  Or if you want/need to replace the stem make sure you buy a stem that is compatible with your fork.

Edit after photo has been added:

I am quite confident you can safely replace these handlebars with "Dutch"-style bike handlebars.  The most complicated would be to make sure you can safely use the brake levers.  Adjust the angle by rotating the lever before tightening it down.  Most brake levers allow to adjust the reach of the lever, in case you have trouble comfortably reaching for them.  If you don't feel comfortable making these adjustments yourself, it would be best to have a bike shop do it for you.  

Good luck!

You didn't provide much information on what handlebar you have now.  Nor what brake levers and shifters are fitted on them.  And this is where the costs might go up quite a bit.  For example, if you currently have drop handle bars, your levers are not going to (safely) fit on the handlebar you want to use.  So then you need to replace the brake levers and shifters, keeping then in mind that you need to buy levers and shifters that are compatible with your current brakes and derailleurs.