Congratulations on your new purchase. I trust it is an exhilarating time for you with lots of projects on the go.
Were you envisaging a small door to access the area as is, or is the more space to be had above, and you wanted a full height cupboard? Either way, you'd look at framing out the opening with structural timber and adding a door jamb around the framed area. You'll likely need to custom make a door which you can do yourself from our internal doors or order something to suit. On the outside, architrave can be used to finish around the opening.
As long as you frame out the opening and connect it to the existing stud work, it should be fine to cut that bracing. Just make sure you screw through it into the stud on the left before cutting it. It's not holding your wall up, but it is providing tension to the framework. That rigidity needs to be compensated for with additional connections and your framework.
After re-reading your post, were you looking to install a shelf niche? That would be a similar process, add some framing and then install the shelving unit. Might I suggest that you use plasterboard and glue it with Liquid Nails to all internal surfaces of the shelves? You can then use plasterboard tape to cover all the joins before plastering them. This will effectively create a very high end looking nook that will appear like it was designed into the property. You should find this step-by-step guide useful as it describes a similar process: How to build a wall niche.