Can I use an LM324 as a buffer amplifier or voltage follower for ADC reading with an ATmega328?
Buffering an ADC input is always a good thing, since they don't handle well source impedence of more than 1k (ADC specs vary, however). The atmega internal ADC is, well, junk like most of the embedded ADCs so the idea is double good.
Now, you didn't say which kind of pressure sensor you are interfacing with. Given the schematic you propose it's not a bridge cell but most probably a voltage output precompensated one. As an example I'll work with an MP3V5004DP since it's the one I use and know. You maybe have a 5V output powered one but the general process is the same.
With sensors, in general, you'll have to cope with different things:

Output impedance; that's easy, it's specified in the datasheet and says how much can you load it (also the main reason for buffering with a voltage follower); NXP in fact doesn't give it but gives a recommended filter cap and a bandwidth so you could calculate that. However with more than 10 megaohm of amplifier impedance that doesn't really matter (it would if you didn't buffer it!)

Signal range, and expecially how much to the rail will it go: that establish the required input common mode range for the amplifier and the output range too. The example sensor, by datasheet, can output from 0.6 to 3V when powered at 3.3V. The LM324 has a CM range from 0 to Vcc-2V so it would need to be powered from at least 5V to cover the whole range. At the output side the situation is mostly the same (5mV to Vcc-1.5V, typically), so it will work, too.

Accuracy you require. This is a complex issue and deserve a whole book to be explained. You have an input in (typically) Pascals and an output in volts. At this point each stage will introduce an error both in error and in offset (at a first approximation), and some calculation is needed to reach a verdict.


For example, the NXP sensor has a ±150mV offset from the nominal 0.6V at 0 bar; also there is a (transduction) gain error of up to ±2.5% (depending on the range you use)
The opamp has its own offset error (3 to 7mV); since we use it as a voltage follower gain error is negligible (otherwise resistor tolerance will enter into play)
The ADC has errors too; the Vref generator has, mostly, since the ADC core itself is digital (usually!). I have no idea if you are converting from VCC, from an internal VRef (most probably) or from an external vref chip. In any case something like 1% of vref error is plausible too.
Looking at the values you could conceivably add them up and devise some mean error or maximum error. However notice how big is the transducer offset tolerance in comparison to the other sources of error: 150mV against some mV of opamp offset. In this case the opamp offset is negligible. In fact NXP recommends some kind of zero routine for it: that would zero the opamp offset too.
As for the gain error: you have 1.5%-2.5% from the transducer, substantially nothing from the opamp and some % from the vref generator. These could amount to, say, about 3% of full scale range error. You could do a full range calibration to reduce it, depending on other conditions.

Last but not least thermal effects: these are already accounted for in the sensor specification. Opamps have drift, however: 7µV/°C typically for the LM324 on the offset, for example. If your enviroment temperature is stable you could ignore them. The Vref has some drift too, it's in the datasheet.

In short: given that transducer, powered at 3.3V and a 5V powered LM324 you should be able to achieve the stated accuracy. If your transducer is instead powered at 5V and you signal is, say, 0.5 to 4.5V (a popular one), the LM324 at 5V is not suitable since it's out of both input CM and output range. You could use a rail-to-rail opamp or simply power the LM324 from 12V, for example.
There are also precision opamp (the LM324 is general purpose) that have really low offsets and/or drifts; of course you pay for these and they usually have some disadvantage (like low bandwidth, usually). For some other kind of transducer they would be more suitable but with a huge starting tolerance like the NXP one it's wasted money.
Remember, it all starts with the spec of the transducer you are using.