COULD is better. and workmate should be colleague.

From just this sentence, it is difficult to tell the context. Also, coworker is another word for workmate. If you look at this sentence in the past, should is acceptable. Here is my argument for should . It could be that you and your coworker were supposed to do a job together but you ended up doing it by yourself. In the end, you regretted doing it by yourself. example = I should not have eaten that food. Most people here are assuming that you are trying to say that you could not have completed the job without your coworker. Depending on what you are trying to say depends on whether you use could or should. If you were trying to say should the "without my workmate" should follow behind. "I should not have managed it without my workmate." If you are trying to say that you were unable to manage it without your coworker than could would be the correct word. hope this makes sense =)

Actually, it is OK here. Don't ask me why, but it really is OK. There are more to modals that ESL books tell you.

Should is not the right word. Should suggests uncertainty and there is no uncertainty here. The statement is a definite statement about a past event. So, the correct statement is "Without my workmate I could not have managed it.". You could also say, "Without my workmate I would not have been able to manage it.".

COULD is the proper word to be used here, because it stresses the capability of doing something. By the other hand, SHOULD is more to express a suggestion. For example: You should go to the doctor when you are sick.