How to keep lemons, oranges and grapefruits fresh longer?
There isn't a whole lot you can do to preserve the actual fruit for what you describe. You could freeze juice in ice cube trays and transfer the cubes into a freezer bag, but even that would lose something compared to squeezing a fresh fruit segment into a drink.

There is one thing I have done for a similar kind of application that you might like. There a huge amount of citrus flavor in the zest of the fruit. Using a vegetable peeler, a sharp knife, a zester or a microplane, remove just the colored part of the peel (the zest), avoid the white pith below, the pith is very bitter. If you have large strips, cut them up tiny before proceeding. If you've used a zester or microplane your zest is already ready for the next step. Mix the zest with perhaps double or triple its volume of sugar, and process in a food processor until the sugar is super fine and no chunks of zest remain. The sugar can now stay in the freezer for months (indefinitely?), it can live on the counter for several days or perhaps much longer. A pinch of that in your drink may just add to your drink what you are looking for.