Can I replace my batter and upgrade to ssd at the same time?

Peter - Take the time to also replace your HD SATA cable as the original cable is not able to support the hight data flow the SSD can offer. Here’s the needed part MacBook Pro 13" Unibody (Mid 2012) Hard Drive Cable and here’s the guide MacBook Pro 13" Unibody Mid 2012 Hard Drive Cable Replacement I also place a strip of electricians tape on the uppercase where the cable crosses over as the rough aluminum tends to wear the cable. You also don’t want to crease the cable around the bends! You want smooth arcs as a sharp bend can damage the thin foil wires within the cable.

You will likely want to get a SATA to USB adapter cable so you can migrate your stuff over StarTech 2.5" SATA to USB adapter

I would setup a bootable OS installer following this guide before you start things How to create a bootable macOS Sierra installer drive and lastly make sure you have free space on your SSD afterwards! SSD’s need breathing space! A 256 GB 1/3 free, 512 GB & 1 TB 1/4 free, 2 TB or larger 1/8 free.

And, yes you can replace the drive and the battery at the same time!