You can do this, you just need a voltage regulator to take the motor's high voltage and provide a lower voltage for the arduino. I cannot rememebr the usual max voltage for an arduino's onboard voltage regulator (if powered via the DC barrel jack), it might be 12V, or maybe 15V? Check for your board, it may vary between exact models. For a 12V battery you're going to have a bit more than 12V when it is freshly charged and not much current is being drawn. The arduino board contains a linear regulator, quite inefficient but ok for the low current an arduino typically uses. If it isn't able to handle the voltage of the batteries, or if you're powering a lot of low voltage sensors as well as the arduino, then buy a switching regulator, pololu sells a lot of them as "step down" DC regulators/converters. It sells a variety with a 5V output, each capable of different maximum currents.
The really important thing here is to put plenty of decoupling capacitors between the arduino's Gnd and 5V rail, this way if the higher voltage supply which is powering the motor drops so severely (like during a sudden start) for a moment that your lower voltage regulator hasn't enough voltage to run, then the big caps (several 100 uF, you can get big SMD ceramics of 100uF each 5V rated, but if you use electrolytics instead then parallel them with atleast a few uF of ceramics) wil keep the arduino and any low current-draw logic powered until the higher voltage rail is back to normal and the regulator can work again. Put these decoupling caps physically close to the arduino.
For current sensing, I would not bother with measuring this for the arduino, the current going there will be minimal compared to that going to the motor. For the motor, if you put a very small, but precise, shunt resistance (like 0.05 ohms, you can buy special resistors with this resistance and a high wattage (therefore current) rating) between the ground of the motor driver (you might be using an ESC module when a brushless motor is in use) and the overall ground of the system (which will also be the ground of the battery), you can use an op-amp to measure the voltage across this small resistor. That voltage signal can be amplified by the op-amp so that your maximum expected current (say 4A) corresponds to a 5V signal coming out of the op-amp and a zero current corresponds to zero volts out of the op-amp. 0.4A in the motor would read around 0.5V in this scanerio. You can read that 0 to 5 V signal with an arduino analog input pin, then on the arduino do the maths to convert it back to a current, and print the result to some sort of character display module. I had a thread recently about op-amps for current sensing, the thing I was after (bi-directional) at the time was a fancy feature you won't need here, but some of the simpler circuits discussed in the thread could be exactly what you'd need.
https://www.eevblog.com/forum/beginners/bidirectional-op-amp-current-measurement-for-shunts-with-both-ends-outside-rails/msg5256327/#msg5256327 (see my image on the first post there)
EDIT: this simulator lets you see what I mean about current monitoring via an op-amp
https://www.falstad.com/circuit/circuitjs.html?ctz=CQAgjCCsB0AM8MUgLAUwLQA4SegJkzAE4B2STTEgZiNskhCtiimcgzDACgqTnMqeKESFh42SCPAhmsHn3DjhoscmXTZXAE7hV6sHnjrZ+LgHNwhtlIPIJUzVQBs2F9kJOQZISUMyZ5jhEniQuQSGYapoA7uFeznGh2HKxRAY4diBpQh4BsZjBGdjZRXlxFPyFkclcqem5bjhgnnI6YrDumJWeFf7trYod8T2F1C2DKYPuglM4XWXt2GOzXZptsL5zzItb-lbaipvLO8vM+wCGK81Z9ddE4AzoYAz9CDLQzZh49AJOzU5UQgQWAfFx4JySKjPZr0TgTeC1Q4+BJiCjDBZ6SQqWBqU6I1FLFF6PGxAxGLGWIy9SZiTa9Mn8eZyZCwKi7Jws2Z8KIyAD6YF5sH532QkEFcHgArAEtIAMw9FZHTwVF5T15eEFvKoXDAJFxkXUjQpEE5ILwIggACVUABnACWNoALucAHYAY1QiJIBopAiEFMmyycQyDkBaXoSEM8wdcYYClpAMcTccgBGT42YnPSsnekC4AHtpDNtshnLr3ggIHhoIIRBs-jRZZAqDzRIwCyA1MWQAAPc5Q9JOfCwJyjviGJzfIjwBggyvSIQgx5EbBdxPQKJcIAmove the resistance up and down to vary the current, see how the output voltage tracks linearly with the changing current. Imagine the variable resistor in the sim represents your motor. The resistor values in this sim are not yet optimised for your use case.