Spirits with cork tops while asthetically pleasing are, to me, always a crap shoot as to how long the cork will survive! I believe there are some types[species] of cork that are perhaps more durable[if you will] than others I don't know if the bottlers check for the highest quality and even if they did - how could you know what you received once its mated to the top!
Having said that - I've been down your road many times in my years of enjoying scotch, cognac and armagnac many of which came with a cork top.
What I've done over the years is save the tops from consumed bottles hoping that I'll have one of the right size to atleast 'plug' the top irrespective of how nice it looks. I've done this for many year hence I happen to have accumulated the supply - for the newcomer its not as easy. Hardware stores sell corks - literally but no pretty top just a tapered cork as one solution after you've strained out the crumbled cork should you choose to do so.
To your other question - DO NOT store a spirit on its side with a cork top the higher alcohol content breaks the cork down. Store cork spirits upright.