Time as we know it

The weirdest thing had me thinking late a few nights ago.

I read somewhere that since the sun, as a star, is past a certain age, each consecutive day is approximately 1 second shorter than the one before it. So, technically, in 86,400 days, we'd miss one whole day. But before that day arrives, 43,200 days into the future, hell, 7,200 days into the future, won't we notice the sun setting at 4 in the evening?

Of course, 1:30 am was too late to be mentally juggling multiples of 3,600 so I fell asleep with a rather confused look on my face.