Upgrading my cockpit- brakes, stem and bars for longer reach
Be aware, if you aren't already, that installing a longer stem will change the steering response - you will need to put in more effort to steer the same amount.

Another way of increasing reach is to change the position of your saddle. If your saddle has rails, then usually you loosen a bolt under the saddle and slide your saddle backwards.

Also, seatposts have a varying amount of layback/setback (the amount by which the saddle is offset), so you could increase reach by finding a seatpost with more layback.

If you don't want to change your seated position, then to counteract the effect of lengthening the stem, you can increase the width of the bars. I'd suggest remaining with flat bars for city riding, unless you want a more aero position. If you stay with flat bars then you can keep the brakes you have, and maybe refresh the cables and blocks for increased performance.

Another reason for sticking with flat bars is that the bike looks fantastic as it is - it would be a shame to change too much about it...