The way to link “just” to “found” is simply to skip (leave out) the “t”. The diphthongs (double vowels) can be difficult. You can practise them separately. Notice the stress is on the preposition, which may be surprising, but is very common, particularly with a phrasal verb. The final “t” is often pronounced not with the tongue, but right down in the throat, where you make a /h/ sound. This sound, which looks a bit like a question mark, is called a glottal stop.
“Find out” means “discover”, so this phrase means “I’ve just discovered/heard” some news or information.
Remember that the -ed past simple verb ending usually has a silent “e”, so it doesn’t add a syllable. Because of the /k/ sound, the “d” becomes unvoiced – /t/. Curiously, the “t” of “it” can be voiced, making it sound like a “d”. This is optional. The final “t” can be pronounced with the tongue, or down where you make a /h/, but isn’t released.
To work something out is to solve a puzzle, mystery or problem.
Notice that the “ed” ending of “worked” doesn’t add a syllable. The “e” is silent, and the “d” is unvoiced, so sounds like a “t”. Also notice the pronunciation of “the” when it’s followed by a vowel. It’s quite different from the dictionary pronunciation.
This phrase means “It ended well”, and you would say it about something you had been worried about, like a difficult exam. My students have exams this week, but I’m sure it will work out OK in the end.
Notice that “I”, when unstressed, can reduce to the short vowel /ʌ/. Because of the “m” of “make”, the “n’t” of “can’t” is pronounced with the lips, not the tongue. There is a stop between the words, but don’t open your lips. Listen and practise. The “h” of “him” is tricky, so we just miss it out.
“Make someone or something out” is a phrasal verb meaning to see or understand clearly. It’s often used in the negative. This phrase means “I can’t understand him well”, or “I don’t know what to think of him.”
We make the link between “last” and “minute” easier by making the t silent. The t at the end of “minute” is pronounced right down in your throat where you make a /h/ sound. We’ve practised this before. This makes it easier to get to the /ʃ/ sound.
This is a noun phrase, used, for example, in a phrase like, “I have to do some last minute shopping.” That means shopping when there’s not much time left. Don’t panic though. Christmas, if you celebrate it, isn’t till Sunday. Plenty of time..