Diane: just to add a bit to what Tom wrote, bleeding the brakes is very simple and takes very little mechanical ability or experience. It just takes one patient friend to help.
First, make sure there is fluid in the reservoir. Alpines are very sensitive to brake fluid type, which is why Tom wanted you to check to see what kind was in there. The recommended type/brand is Girling, which is still available at many auto parts stores. However, if the previous owner put something different in, then it would be good to find out first. If that isn't possible, I am sure people here can guide you through your options better than I can.
If there is fluid in the reservoir, then Tom is hopefully correct about how the air got into the system. Getting the air out is as simple as having one of you pump the brake pedal until you get pressure (if you don't get pressure and there is fluid in the reservoir, then other work will be needed) and then holding down the pedal while the other person opens up the bleed nipple on the right rear drum brake. (The workshop manual, which can be found here,
http://www.sunbeamalpine.org/index.php?categoryid=89, will show you where everything is.) You don't want to just let the fluid run into the ground, as it is toxic and also a pretty effective paint remover, so you need to run a vinyl tube from the bleed nipple to a jar. The pedal will go soft as soon as you open the nipple, but the "driver" needs to keep his/her foot on the pedal until the bleed screw is closed. The process is repeated until there is no more air coming out of the system and then moved to the right front disk brake and then the left front. As long as you remember to keep the fluid topped up (letting it run out lets air back in, but just requires you to start over) it is really simple and only takes a few minutes. Be careful not to over tighten the bleed screws, especially in the back, and you are good to go! Or, more accurately, to stop.
Now, if it loses pressure again and the pedal feels spongy again after that, it means that air is getting into the system from a leak somewhere and that needs to be fixed immediately, least you lose the brakes when you most need them.