In everyday speech, notre reduces most of the time to /nɔt/ when the following word begins with a consonant.
Petit has a e-muet/schwa (/ə/) in its initial syllable which is notoriously unstable and may or may not be pronounced depending on speaker habits or their dialect, speech conditions, the surrounding sounds, etc. Petit /pəti/ can thus also be realised as /pti/ or even as /ti/. Normally this reduction is seriously inhibited if the preceding context is a word ending in a consonant, like "not'", but this isn't absolute, especially with very common words like petit, peut-être, pronouns like "je" or determiners like le or ce.
Put both of those together, and you have a pronunciation of "et notre petit dernier" as "et not' p'tit dernier" /e nɔt pti dɛrnje/. The /p/ is hard to make out, but feels present to me. In a Praat spectogram, there is noise appearing between the closure of the /t/ of notre and the release of the /t/ of "tit".
Limiting the jargon to the maximum, when you pronounce a sound like /t/, you block the airstream with your tongue somewhere between the teeth and the domed part of the palate, let the pressure build up a bit then release it with a burst. When two /t/ are adjacent, as in netteté (/nɛtte/), the pressure build up phase is held for roughly twice as long, but the closure and burst phases happen only once.
What the speaker seems to have done here is block his airstream when saying the /t/ of notre and let this closure remain in place until the /t/ of petit, at which point he released the air blocked behind his teeth, as if producing a long /tt/. But in the middle of this closure phase, he closed then opened his lips as if pronouncing /p/. Because the airstream was already blocked behind the lips, there was no build up of air and thus produced no burst, but still modified the soundwave a bit, which helps identify this as /nɔtpti/ even if it's not very audible as a distinct segment.