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http://mathr.co.uk/blog/2013-04-01_interior_coordinates_in_the_mandelbrot_set.html gives an image where the interior is colored according to the angle from the base of a bulb.

If there was some easy means of determining your angle in a bulb like this it would tremendously enhance the ability of |Z'| to predict the real cycle period for that location. But I can't really follow the math well enough. Is there a relatively cheap way to determine what your angle is relative to the center and base of a bulb?

Unfortunately the maths works the other way - you need to know the period to find the interior coordinates.  What I did in that blog post was to try with each period from 1 upwards and see if the result is sensible (distance less than 1). 

There might be a way to optimize this search using "partials" - the idea being to test only those periods that occur in the partials list - though I haven't checked that it works for this particular use-case.

"Partials" is my term for the set of n with 0 < n such that |z_n| < |z_m| for all 0 < m < n, where z_{n+1} = z_n^2 + c, z_0 = 0.  This is related to "atom period domain" colouring: http://mrob.com/pub/muency/atomdomain.html

A few examples:

-9.82053364278e-2 +
0.87751161636 i
is the center of a bulb with period 30 and approximate size 8.7e-4
its partials are 1 3 6 12 30

-1.40107054826248 +
1.68078322683058e-2 i
is the center of a cardioid with period 25 and approximate size 2.9e-7
its partials are 1 2 4 8 25

-6.30751837180080329933379814864882594423413629277790243935409e-02 +
9.97813152226579778761450011018468925066022924931316287706002e-01 i
is the center of a cardioid with period 660 and approximate size 1e-51
its partials are 1 3 4 5 12 34 133 660

-1.949583466095265215576424927006606703613013182307914337344552997126238598475224082315026579e+00 + -7.76868505224924928703440073040924718407938044210292978384443422263629366944037056031909997e-06 i
is the center of a cardioid with period 1820 and approximate size 3.5e-83
its partials are 1 2 3 4 9 25 83 359 1820