The '87 Spider (as well as all series 3 and even series 4 Spiders) is slow off the line. Once you get moving, then it's fine. In the series 3 Spider, the power has to do with how Alfa (as well as most car manufacturers of the period) limited the engine output in order to reduce emissions at the tail pipe and meet what were then new regulations. Hence, there is actually much more power to be had, but it will take some doing.
You cannot upgrade software (chip) in a Bosch L-Jetronic ECU, since there isn't any. (You can with the Bosch Motronic that Alfa starting fitting to the Spiders in 1990.) Going with a turbo is in fact the most cost effective way to increase power, but $2k won't be enough to finance such a set-up. In the eighties Shankle Engineering sold a kit for the Spiders, and the cost was $3500. Today, you would probably have to budget at least $5k to do such a set-up, and that's assuming that you will be doing some of the work yourself. If you pay a pro to do it for you, then it will cost more, since custom work is charged by the hour.
You can also increase power in a conventional manner by fitting a free flow exhaust, a free flow catalytic converter, and headers. Other upgrades that could follow could be a ported/polished head (there are different levels of upgrades here), higher compression pistons, high lift camshafts, and a modified ignition. This, as you pointed out, is not inexpensive, but it is effective. Please note that there isn't one single upgrade that will yield a "wow effect". Instead, every little bit will help and you can get to a point where your Spider could be significantly faster off the line, and going uphill, than a stock example. (One of my Spiders in fact has been fitted with upgrades such as the ones I described, and it is very fast off the line, and going uphill. It's a blast to drive.)
I think the best suggestion I could make is for you to lay out a detailed plan about everything you want to do related to power increase, with realistic $$ figures, and then follow it. That is not just the most time efficient (no $$ waisted) and cost effective approach, but it also allows you to firm up and/or refine your goal on paper and even drive other cars that have similar mods, before doing anything to yours. You may decide that even if you don't have the money now, it could be worth waiting for your finances to be in place in order to proceed with your ultimate set-up.