As you have said that the person who worked on the project before is well liked and respected within the company, while you are the new guy with no good will or social capital built up in the organisation, the last thing you should be doing is forming any kind of criticism of the code or the person who wrote it.
However, if the project is truly "Broken", then the person who worked on it before will not be the untouchable God type. He may be the asshole programmer from Hell who purposefully wrote code that could not be maintained by anyone else, in which case you are his patsy and you are probably on a losing bet, because he is just waiting in the wings to swoop in, pull an all-nighter bashing away at the keyboard to rescue the company from the incompetence of the new guy, all for a measly 50% pay rise... I have seen it happen to a few contractors and it is not pretty.
The "obvious" solution, of course, is to quit and find another gig. However, the next place (or the one after that) will probably have a similar scenario, so the best approach would be to start learning to tackle the problem on this project.
That means that step 1 is to look at where YOU went wrong. By that, I mean that either your initial code analysis was incomplete (you did not check out the code before taking the assignment, or maybe you had no opportunity to check it out), or you started coding before understanding the existing structure (or lack of). Yes, you are under pressure to add value, contribute, justify your existence, and so on... but that will be doubly true next month. If you cannot make the argument in the first week/month that you need to review the existing code before making changes and adding features, then you are not going to be able to make that argument at any point. It takes a particular kind of coder to write updates to existing code without first understanding what the existing stuff does, and that type is rare, especially when dealing with an un-maintainable bird's nest of code.
If the code is already documented, verify that the documentation is accurate. If it is not already documented, then document it. The code may need to be re-factored before you can make any meaningful contribution, but right at the beginning of the project is the only possible window you will have for that kind of analysis.
If your response at this point is "I do not have time to document the code", then my advice would be to leave with your sanity and most of your reputation intact. You have already seen that coding hot, without any insight into what you are working with, causes unexpected and unexplained problems.