Sample Rate Conversion between 32K and 44.1K
The standard way of doing this is a poly phase filter. Unfortunately it's not straight forward and the devil is in the details.


You need to determine cutoff frequency, number of phases, length of filter, and (to a lesser extent) shape of filter
These are typically derived from your specific requirements: what's your allowable bandpass ripple, stop band attenuation, residual aliasing, sine wave modulation, up to what frequency, etc. 
How exact does your final sample rate need to be? If you can tolerate a quarter percent of frequency error you can approximate the ratio as 8/11. If it needs to be better than that, you may just as well go with an asynchronous converter
MIPS and memory footprint depend very much different factors. Number of channels to convert,  $C$, filter length , $N$, and number of phases $P$. Storage requirements are roughly $(P+C) \cdot N$. MIPS for synchronous are $C \cdot N$ multiply/adds plus some overhead for phase indexing. 
Asynchronous will be more, maybe $(C+2) \cdot N$, depending on how exactly phase interpolation is done. 
It works well with fixed point