Just Bento

I have tried dry ume tea in a few Japanese hotel rooms where it is sometimes given as tea bags. It was nice enough but I find I prefer 'regular' teas better (hojicha, sencha, genmaicha etc.)
Much preferred is the 'wet tea' made from ume and honey which can be bought in Jars from Korean companies. It's similar to the jelly/jam-like yuzu/yuja tea which I'm also extremely fond of.
There's a photo of the brand I use at the top of this page labelled as Honey Plum Tea
http://www.samhwatea.co.kr/japanese/product/index_1.asp

I like alcohol flavoured with ume (I've had many kinds and loved them all) and I've also enjoyed it mixed with soda water... although these were particularly nice gift ume that were preserved in honey and it was the ume infused honey that made the sodawater taste so nice. There are quite a few varieties of soft drinks with ume flavouring on the market.

As for cooking
I add ume to ochazuke.
I mix it with pickled beetroot and mash them together before blending with a combination of mayonnaise and creme fraiche/soured cream to make a gorgeous pink dressing for things like fish and new potatoes.
If you are cooking fish in stock or water (eg for soup) adding an ume to the cooking liquid is supposed to keep the fish from flaking and breaking so easily.

And, of course, there are all the very many ume candy varieties. Some with a refreshing ume flavour, others are intensely sour.
I have to admit that I've never met anybody who thought ume was gross.

I have only tried umeboshi once, and they were too salty for me. I love lemons too, but if those umeboshi were typical then my favourite thing to do with them is feed them to someone else!