Cope with -- implies that the problem is so big that they can't handle it well. Typically you hear something like "they are doing their best to cope with the problem" which implies they aren't doing as good of a job as would be wished for.
deal with / handle are very similar and imply the problem is being resolved competently.
tackle -- typically means that someone is going to try to aggressively, try to solve the problem. Might be worth watching a video clip of a tacle in rugby to get an idea of where this comes from. Also there is no certainty that the problem will be resolved.