Inverting amplifier to measure shunt voltage?
You can get op-amps that are dc accurate down to a handful of micro volts so there's no problem there but, given that the current can be negative as well as positive I'd create a small negative supply to power the op-amp symmetrically about the ground point. Then I'd go for something like an OPA333 as the op-amp. If you need more bandwidth then something like an ADA4528 is also good.

If you are looking for budget parts - you get what you pay for.

Basically run the op-amp as a non-inverting amplifier and add offset to centralize the signal produced into the range of your microprocessor's ADC.

You might even consider a low offset-voltage Instrumentation Amplifier like the AD8222 fed from a symmetrical supply - it has a pin you can set for the offset called "REF".

You can get very cheap DC-to DC converters (very small too) that can be wired to take a positive supply voltage and produce a stable negative output voltage.