Cocktail lime juice
There is no substitute for fresh lime juice. I have tried all the little green plastic bottles, the big glass bottles, freezing fresh lime juice, frozen concentrated limeade (in my younger years), and lots of other substitutes. But in a drink, especially a cocktail in an up glass, fresh lime juice has no adequate substitute.
Especially not Rose's Lime Juice! Don't go near that stuff as it is to fresh lime juice as TGI Friday's Mai Tai is to Trader Vic's original. Rose's Lime Juice is sweetened lime juice with citric acid, sodium metabisulfite, you get the picture.
So you need to squeeze your own fruit. The good news is that limes last a pretty good long time in the refrigerator and they are pretty cheap. I have gotten two weeks or more out of my limes. Sometimes you will start to see some browning on the outer skin. As long as this hasn't penetrated to the inside, you can still squeeze lime juice from it and you'll be OK.
Just get a hand squeezer and you'll be fine. I have used a give away citrus press that came with a bottle of Tequila with no problems for years. You can store fresh lime juice for about a day as long as you keep it cold.
Drinks with lime juice
Lime juice has a long history in mixed drinks. It was used on a wide scale as an antiscorbutic (scurvy preventative) in the Royal Navy starting in the 17th Century. Lemons actually have much more Vitamin C than limes but England had better access to limes in her West Indies colonies. Later other nations figured this out but the Royal Navy started the trend and therefore their sailors have forever been called "limeys".
Of course, those sailors started to mix their lime juice with rum and bequeathed grog upon the world. Colonel Joe Rickey gave us his namesake drink. Fresh lime juice is also a key ingredient in many Tiki drinks and in drinks that originated in the tropics where limes are plentiful. Daiquiris and Margaritas are two huge examples.
Do you have any storage tips?