Shukracharya apparently asked that the Asuras to be compensated for being cheated out of Amrita and after his austerities Shiva gave him the mrita sanjivani mantra. Is it the same as the Mahamrityunjaya mantra?
No it is not the same MahAmrityunjaya Mantra. It is a special Mantra which is formed by combining the GAyatri Mantra and the Mrityunjaya Mantra in a certain manner. It has a special name viz- "ShukropAsita Mrityunjaya Mantra" Or "the Mrityunjaya Mantra which was worshipped/practiced/mastered by Shukra"
Quoting from the Krishnachandra AgamvAgis's TantraSAra's pp 505:
atha shukropAsita mrityunjaya mantrah ||
gAyatri prathamam pAdam tryambakapAdyeikam tathA |
gAyatri dvitiyam pAdam tryambakadvitiyam pAdam |
gAyatritritiyam pAdam tryambaka sheshapAdam ||
mantro tathA - Om tatsavitur varenyam tryambkam yajAmahe sugandhim pushtivardhanam vargodevasya ... ||
The Mantra is thus formed by combining the GAyatri Mantra and the Tryambaka Mantra as follows:
First part of GAyatri is added first followed by the first part of the Tryambaka Mantra. Thereafter, GAyatri Mantra's second part is added and followed by Tryambaka's second part. Then added is GAytri Mantra's third part and similarly followed by Tryambaka's third part.
It is now easy to see what the Mantra turns out to be finally. Not giving in full but it is like this " Om tatsavitur varenyam tryambkam yajAmahe sugandhim pushtivardhanam vargodevasya .... ||"
The way/technique this Mantra is formed (by prefixing/suffixing Mantras to another Mantra) is known in the Tantras as Samputana.
Then the text gives the DhyAnam for Tryamabaka MahArudra, who is to be worshipped and the Japa is thereafter to be done.
Evam dhyAtvAvAhya tryambakAya mahArudrAya namah ityanena pujayet |
asya japAt sarvasiddhirbhavati ||
This passage says that Puja Mantra to be used in the Prayoga is ""tryambka mahArudrAya namaha". And, it also says that chanting of this Mantra bestows all sorts of fulfillments.
However, note that, there is another Mantra called the Mritasanjivani Mantra dedicated to Lord Shiva. It is also derived from the Tryambaka Mantra by adding a prefix (X) both before and after the usual Tryambaka Mantra.
The Prefix X is- a combination of the "Bhur Bhuvah Swah"+"Joom Sah" (not giving the exact Mantra though).
But, it is not mentioned in the text whether Shukra worshipped it or not.