Is there a part of the brain that controls blood pressure?

No. There is no certain brain part that controls blood pressure. Blood pressure control is multifacet mechanism but is primarily regulated by so called "baroreceptors" in the vessel wall. Those receptors are able to detect the degree of the vessel wall stretch and react to it accordingly. This triggers a whole cascade of other mechanisms of BP regulation.
Brain controls all. You see blood, you turn pale, and you faint. You see someone you hate, you flush, your heart pounds in your chest, and your head pulsates. Your blood pressure went to near zero in the first and skyrocketed in the second setting. The automatic or autonomic function of the body is very much controlled by complex centers and systems within tne brain. No one spot is blood pressure center.
Most definitely. The fantastically named medulla oblongata (or "medulla" for short) modulates basic body functions like this. In response to sensors staggered throughout the body, and in association with other organs (most notably the kidneys), it controls secretion of Epinephrine and norephinephrine to help modulate blood pressure.