Looks like you probably nailed it, if you want some assurance look for numbers on the carburetor and do a google search. As an example heres a pic of my hedge trimmers carb showing some numbers/letters. If yours is a good name brand (Zama, Mikuni ect) I'd rebuild it with parts from the one you've shown us if possible, cheap carbs often give alot of trouble due to poor castings and QC and need to be worked over (just my experiences).
What you are planning to do with the motor is important in deciding both carb size and what style of air filter to use, if it's going to be in dirty/dusty/muddy areas you'll want a factory style that has it's own airbox, mostly dry grass/pavement and an open air filter would be preffered. More pics of the motor would also help. Is it a 2 or 4 stroke engine? Heres an example of some air filter, for the top 2 youd need an adapter which shouldn't be hard to make
You want to find something like a round cap that the filter fits on (pvc pipe cap maybe?) then cut/drill holes so it bolts on with the carb screws. If you're just bombing around I wouldn't get too hung up on carb size or spending alot of money on a filter, keep it cheap and simple and have fun!