I use "Firefox ESR" on "Precise Light" and with only a few tabs open it is under 512MB. This will quickly be exceeded as you open more tabs. However, I think your main difficulty may be your CPU is too slow to browse many(most) websites or play anything but the lowest resolution videos <= 360p? "Precise Light" can do some other things quite well, basic text processing, spreadsheets, ... but probably not web browsing or higher resolution videos - really depends on exactly how fast your P4 is. In my experience most activities in Puppy are CPU bound not memory bound If you have a later higher end P4 then even with only 512MB of RAM (and swap space on a hard disk) I think Puppy will do much better then I stated.
This is my seat-of-the-pants thinking - if you run the following sysbench test and get a score significantly greater than 10 seconds your machine will struggle mightly (i.e., be unusably slow) to browse modern web sites, play YouTube videos or higher resolution videos.
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sysbench --test=cpu --cpu-max-prime=10000 --num-threads=2 run
--num-threads should match the number of cores.
Also, On "Precise Light" I was never able to get "Firefox ESR" and a video player to work together - after installing the video player the audio stopped working in Firefox. I suspect that may be the reason "Precise Light" does not have a video player pre-installed.
The current version of "Precise Light" is 5.7.2. Also, using Rufus or "Universal USB Installer" on the "Precise Light" iso makes an unbootable flash drive. There is an easy workaround which I can give you if you decide to use "Precise Light". This is really a shame as "Precise Light" is quite a good distribution but I think many users will really be put off when the flash drive does not boot.
If you are not totally wedded to Ubuntu I think "Dpup Stretch" (Debian based) would be a better experience. I don't use it because I am totally wedded to Ubuntu based Puppies but when I tried it I was very impressed.
And always check that the firewall is enabled - I don't know how this happened but on at least one installation of Puppy my firewall was off, I probably made a mistake somewhere but now I always verify the firewall is actually on.
ADDENDUM
I just did a session on "Precise Light" browsing with "Firefox ESR" with only two tabs open (one YouTube and the other I don't remember) but the memory usage was ~ 490MB. I think you will need swap space but I think if you have a fast enough CPU it may still work out. But, I don't think you will be able to have many tabs open - a tab these days seem to use ~ 100MB.