The flowers normally dry out and fall off when they are done, so what's happening at the top of the taller stem is perfectly normal, provided there aren't any bugs on it... hard to tell from the photos. The flower stems are sticky so they might just have some dust or debris stuck on them, but you should keep an eye out for bugs that might look like this
and if that's what you've got, cut off all the affected inflorescences at the base. Be careful not to touch the sap that will ooze from the wound.
As the older flowers fall off in old age, newer ones will be generated, so most likely the plant will be in near-continuous bloom for the next few months, in good light. That's one virtue of the crown of thorns. The leaves will also die off as new ones are being generated (these plants do make a fair amount of leaf litter over time) so as long as the end of the stem stays as leafy as it is, there should be no problem.
Provide the most light you can indoors, like right by your sunniest window. Continue to water deeply like you are doing, maybe every week or so if the plant is getting good light indoors. You might consider care instructions as part of the gift, to make sure it keeps on giving.
The plant is not cold-hardy (so the recipient knows, when winter comes) but it does enjoy a fair amount of outdoor sun during the warmer months. Just be careful if you move the plant outside for the season... make the transition from indoor sun to outdoor sun gradual, starting in bright shade for a couple of weeks before a bit of morning sun, and then only after a few weeks of adjustment direct overhead sun.