This is a very generic question. Your competitor have blasted their site and you though process is to outrank them by using quality links. The theory says that this may work, but personally I would never say reliably that this will work 100% of the times. Blasts sometimes lead to weird result for the good and for the bad. We have seen plenty of times in this forum, blasted pages ranking first for a good period of time.
I'm the no1 detractor of competing for keywords with other sites. My SEO philosophy is mass kw building disregarding competitors after kw research (except for link building guidance and tips). Meaning that if you can check the linking profile of your competitor and spotting high DR links that are relatively easy to be adquired by yourself, do so. Otherwise, keep on working with other good quality links from other sources, like PR, PBN links and maybe, if you were not as lazy as you said in the first message, building a little bit of a tier 1.