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I know the Arabic definite article /l/ is a common example of this, as it assimilates to the following consonant if it's coronal. This would probably be easier in auto-segmental notation but how would you write the rule in feature notation? I've heard about α place, is there also an α manner or is that not a feature that's used?
Extra: for reduplication rules, numbers have been used to indicate a copying sound (like 1 2 --> 1 1 2). Would this same method work for assimilation?
The answer depends on your metatheory. In standard feature theory, there is no feature "place" or "manner", so "αplace" or "αplace,βmanner" is meaningless. However, that doesn't stop people from writing rules like that: essentially what they do is redefine the notation and redefine what a feature is (I disapprove, but it happens). The basic problem is that in the linear theory, there isn't anything that refers to classes of features, so you'd have to specifically list all of the changing features, and the rule gets to be pretty big. So this is one of the advantages of autosegmental theory.
This topic actually comes up here, somewhat, in a discussion of what the right formal account of assimilations is. It's clear that we need some mechanism for talking about classes of features, like "place", but there is more than one way to actually get that -- and there is an alternative (first hinted at by McCawley decades ago) to the SPE feature variable notation.
[Clarification] A feature is understood to be a well-defined acoustic state or articulatory action, which either is or is not. Neither "place" nor "voice" satisfy that requirement, so "place" can't be a feature. Ladefoged did propose multi-valued features that would include place, but those were features in name only, since at the time the only concept that was available for representations was "feature".
[On numbers] Use of the numeric transformational format as found in early statements of reduplication is another possibility for total assimilation, if and only if your theory allows such unrestricted rewrite rules. Phonologists generally hate them. The general format for Arabic-like rules would be
[+lateral] [+coronal]
1 2 → 2 2
The notation also could be used for partial assimilation, e.g. place only, by adding the non-changing features to the target element, e.g.
[+cons,αvoi,βnasal,γlat...] [+consonantal] [αvoi,βnasal,γlat]
1 2 → 2 2
which means "make the copy of term 2 have the features of voicing, nasality, lateralism... that input term one had"
I don't know of an α manner convention, but it would surprise me if no one had ever proposed one. So many people confuse phonological notation with phonological theory! If all you want is an easy way to say it, just say "the l assimilates completely". That's easy. If you want a phonological argument, take a look at the SPE argument for α place and see whether a parallel argument can be constructed for α manner. That would be doing old-style generative phonology.
Another theoretical approach is to adopt Stampe's Natural Phonology in which "ease" is interpreted as lack of constraints on the universal phonological processes. Then every respect in which the l fails to assimilate requires a constraint that prevents an assimilatory process from applying, so the easiest assimilation is the complete assimilation that we find for Arabic l. You get complete assimilation in the absence of any constraint preventing a feature from assimilating.
