As of right now, there are approximately 585 quests in the entirety of the MSQ, beginning from your very first quest at level 1 and ending with the most recent level 70 quests in patch 4.56. Of those quests, approximately 285 take place before patch 3.0. This means that you have somewhere around 300 quests left until the end of the MSQ, if you just began Heavensward.

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So, let's do some math. Right now, we're about a month out from ShB early access almost to the day, so let's use the nice round number of 30. You could power through the MSQ in chunks over a few days or weeks if you really wanted to, but since you have so much time, you could also do just ten quests per day and get caught up just in time to join us at launch.

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In other words, it's totally possible you could catch up to the current MSQ. The biggest roadblock you might face is actually ilvl requirements during the 3.X and 4.X MSQ instances, so I'd recommend you try to do a little more than 10 quests per day just to be certain, but you have plenty of time to spare even if you only have an hour or two to dedicate to the game per day. Hope this helps!