1Please read the entire entry including the notes at the bottom before downvoting this answer! I don't like this answer in principle but the tag RAW requires this answer that I strongly disagree with in implementation!
No, going strictly Rules-As-Written2 a Demilich can NOT use Astral Projection on itself (and by extension can NOT use the ability at all.3)
Extra Unintended Consequences: Not only can the Demilich NOT use Astral Projection on itself, it can't use numerous other self buffing spells from it's Wizard's spell list. However, it actually can use it's at will Harm ability! It sounds contradictory but is explained below4
IMHO, Such a strict & narrow reading of the rules weakens the creature so much one would have to ask what Lich would ever seek to "progress" to this level! Surely with these restrictions placed on it it wouldn't be worthy of a CR29?5
So, if I dislike this answer so much and it seems so contradictory, where do I get the data for this answer? And why is it so difficult to get a straight answer? Well, since it's never specifically & independently defined, we get into the several rule issues such as: "Specific takes precedence over general rules" & "Primary Source" specific which might get murky when dealing with mixing 3.0 & 3.5.
On to the "Meat & Potatoes"
To get our answer first we have to look for the definition of "Magic Immunity."
The first & only place it is defined is in the "Rules Compendium." While this is not a "Primary Source" when it comes to deciding which book takes precedence during a rules conflict, since there are no other definitions, let alone any that conflict, that doesn't matter and it's ruling stands. Furthermore, we do see it in the 3.0 -> 3.5 conversion guides listed several times & each time it is giving the exact same definition as this:
(Creature) is immune to any spell or spell-like ability that allows spell resistance. In addition, certain spells and effects function differently against the creature, as noted in its entry.
4 Since the "Immune" block specifically says that "certain spells & effects function...as noted in it's stat block" this allows it's Harm ability to be used as it's stat block specifically mentions that it uses this ability to "heal" itself. At exactly the same time it makes no mention in it's stat block as how the Demilich is to make use of it's Astral Projection SLA, which as we've already established requires the caster to be the target, makes the SLA unusable.
From this updated definition, we can see that "Magic Immunity" was updated to be defined as Immunity ONLY against spells with Spell Resistance. However, it was NOT updated to be defined as "Spell Immunity". This is a Very important distinction!
Why is it such an important distinction? Because "Generally" speaking the rules state:
A creature that has immunity to an effect is never harmed (or helped) by that effect. A creature cannot suppress an immunity in > order to receive a beneficial effect. emphasis mine
This means that the creature can't benefit from any beneficial spell it is immune to, even if it were to cast it itself! However, Spell Immunity has "Specific Rules" which over-rule the "general rule" explicitly allowing the creature the option/ability to receive these benefits. Neither of these "Specific Rules" for Spell Immunity comes into play for our Demilich because his Immunity was specifically NOT redefined to do so. (in order to avoid confusion, I left out the referenced rules as to avoid unnecessary confusion in an already lengthy and somewhat convoluted answer, however, the links to the abilities and my other answer both go into detail)
Bottom Line: Rules-As-Written prevent the Demilich from using it's Astral Projection
1 I generally don't like to post multiple answers, especially contradicting myself. However, I can't include this information and make the previous answer, or this answer coherent, they simply are at odds with eachother & as such requires a separate answer.
2 While this answer is definitely 1 interpretation of the RAW, I believe that considering the conversion from 3.0 -> 3.5 & all the erratas that these rules in-particular are written along a much larger grey line than most others. That said, I think this is a solid answer but would still recommend sticking with my previous answer ruling in the opposite.
3 When reading the spell, Astral Projection we can see the the spell requires the person casting it to be the primary target, with the option of adding additional "travelers" as desired, up to the limit based on CL. So if "You" are no longer a valid target then sadly, nobody can go.
5Final Note If going 100% RAW & ruling the Demilich can't cast this (Astral Projection) or any of it's own spells on itself I think you have to rework the CR. There is simply no way that a Demilich who can't cast many of it's own "Self Buff" spells (whether from it's own spell casting or magic item) can be a CR29! (the list of spells that allow spell resistance is quite expansive, too much to type here, and this list only grows with every splat book added)