This one is difficult to explain, but let me try! So, as Tom said, the correct one in your first pair is "there comes a time". To make it simple and grammatically consistent, let's change it to "There will come a time when you will beg me for help." Why "a time" and not "time" or "the time"? In this case "a time" refers to a time period or a moment. It could be substituted with "a moment" or "a period". Because it hasn't come yet, it is still undefined/undetermined, so we use the non-specific article "a". In the second pair, the correct one is "When it comes time for me to..." Why "time" and not "a time" or "the time"? So first of all "comes" might sound a little funny here. You could easily substitute "becomes" and it might sound clearer and means the same thing. In this case, the "time" referred to is defined. "Time (for me) to [for example] ("die"). So because it is something defined, we should probably use the specific article "the". But we don't! In this case, there is a set pattern that maybe is more important than the usual pattern: In expressions with "time to [verb]" we usually don't use "the". It makes sense grammatically to use "the", but we don't. Some examples: It's time to go to school. It is now time to wake up. * There was no time to clean the house. Do you have time to eat breakfast? When is it going to be time to eat mochi?" * If you put the "time" part at the beginning of the sentence, then you should use "the": "The time to wake up is now."

Usually, "time" is general while "a time" one specific time. But I don't think these rules apply to the expressions you have shown which don't change. The expression is "There comes a time", never "There comes time". Similarly, the expression is "When it comes time" not "When it comes a time". I hope that helps.