Yes, the goal is achievable... if you plan a path to achieve it. ;-)
I can think of a couple of reasons why you'd feel tired. Technique is of course one of them. Pace is another, and so to is both your breathing, and your expenditure of energy, which basically comes down to how used to the exercise you are. If you can run for an hour, but can't skip, it's because you're using your body in a way it isn't used to, and your nervous system is signalling the brain in ways that the body isn't used to.
First things first... don't try to achieve the whole goal at once. Pace yourself, and build yourself up to be able to meet the task. build a plan to achieve the goal in stages.
How long can you skip for now? If you slow your pace slightly, will you be able to skip for longer? If you control your breathing, in through mouth, out through nose, and try to slow the exhale, does this change the time you can skip for?
When you know what your present baseline is, then create a plan to extend your time. This is as much about developing your cardio as it is about developing the way you think and train to achieve the goal you have set. This is also effectively how you tackle training for a 5k... or more...!
You could start out training in simple intervals. Skip just below your maximum effort for a set time (less than reaching exhaustion). Then skip at your maximum effort for a set time (less than before, but more than is strictly comfortable). and do this pattern of easy and hard with rests in between. For example easy skipping for 30 seconds, rest for a minute, hard skipping for 15-20 seconds, rest for two minutes, and repeat for as long as you wish... perhaps 20-30 minutes worth. Do this for the first week. The next week, increase the times of your skipping by 10 seconds for each interval, but maintain the rest periods. The following week, increase the skipping times again and start to reduce the hard skipping times and the rest after the hard skipping until you've removed the hard skipping intervals and are just doing the easier intervals. After this, start to gradually reduce the rest intervals until you are skipping longer, and resting less.
Another simpler formula would be to skip for as long as you can, then rest for the same amount of time as you can skip, but skip at a slightly slower pace than you've been using, and repeat for 30-60 minutes. Each week, increase your skipping time by a small amount, and decrease the rest times by a small amount until you can skip for the 30 minutes you are targeting. Once you've achieved your time, then you push yourself to achieve a performance goal... faster skipping, or for longer... skipping while weighted, and so on... whatever you decide will help you to achieve the next fitness goal you've decided to aim for through skipping.
Another thing to consider, is that when you feel drained of energy all of a sudden, you are hitting a performance wall as your body changes from using one energy system to another. I won't go into the differences between aerobic, ATP and Glcolytic systems, however it IS interesting that we have a delay when switching from one system to another, where we feel drained of energy and just want to stop going. This delay can be a matter of seconds to a minute or two. Knowing this, if you can push through this wall, you can continue to perform, albeit at a less efficient pace as your body tries to adjust to the need to draw energy from an alternate source. Efficient technique helps here, moving muscles minimally to achieve the movement required... smaller hops and smaller arm movements, using mostly wrist and ankle if that makes sense without a video to show what I mean. This is where weekly modified interval training can help, as you learn to push against the wall, and through training you delay the onset of the wall... or you train yourself to overcome it if you train for endurance over time.