On some IRC channels such as #bitcoin-core-dev there are third party managed conversation logs so you can follow the conversation from when you were offline without setting up a bouncer. However, I don’t think (at the time of writing, September 2020) that there are conversation logs for the #bitcoin-core-pr-reviews channel except for the weekly meeting which is logged on the Bitcoin Core PR review club site. You definitely can’t receive messages whilst you are offline.
There are various options to set up a bouncer but in the interests of simplicity I will just outline one assuming you followed the instructions here to install Textual on MacOS.
You will need a virtual private server (VPS) to be running 24/7 to log the discussion and receive messages on your behalf and then you can connect to that server whenever you come back online. This will require a monthly subscription.
So your first step is to choose a VPS provider. One easy to use option is Digital Ocean but there are others that perform the same function and will be equivalently priced. (We generally avoid product recommendations on this site so please excuse this counterexample.) You can connect directly to your VPS/Digital Ocean Droplet using the instructions here. There is a tutorial for setting up SSH keys here.
Once you have set up your VPS you will need to install ZNC on that server. Instructions for doing that are here.
Instructions for connecting your Textual client to the ZNC bouncer are here.