Out of Pod Experience

Your CPU should be quite sufficient.
Your GPU, well, it should be able to run it, but not smoothly.
Even in the lowest possible graphic detail and with a modest resolution, also with lowest quality // highest performance driver settings, I wouldn't expect more than 15 FPS, if even that much.
The 8400 GS was the bottom of the retail entry-level NVIDIA video card lineup 5 years ago, so no wonder it's pretty crappy today.

I would highly recommend buying something like, say, a 1GB Radeon HD 6570 (for about 55$, which is more or less the price of GW2 ).
The 6570 is about 5 to 6 times faster than the card you now have and uses up about as much power (44W vs 40W) so you would not need to worry about power source issues or anything like that.
With it, you should be able to run it in medium-to-highish graphic detail at over 30 FPS almost everywhere.

Alternatively, you could go with a 1GB GeForce GT 630 - slightly higher price (starts at about 60$), slightly worse performance (about 10% slower on average), higher power usage (65W) - if you insist on sticking with NVIDIA (they kinda' suck at the budget price levels).
You might also need to make sure the PSU can handle it - there's not that much extra power draw, still can be serviced by the PCI-E slot alone (no need for PCI-E power connectors), but the PSU must have some small reserve left over for it, after accounting for PSU aging (a mediocre quality PSU can lose anywhere up to almost 10% of its rated "fresh" power per year of usage, especially if it's been used near its current power limit, so the deterioration accelerates with age usually).

If you have some money to spare though, the best price/quality buy of the moment would probably be a 1GB Radeon HD 7750 (starting at around 95$).
It uses up ~55W, but it's noticeably more than twice as fast as any of the other two cards mentioned above (not quite three times faster, but close).

The cheapest Radeon HD 6570 are usually low-profile cards, so they should fit in a flat desktop machine.
Haven't seen any low profile GT 630 though, but some might exist.
I have seen at least one Radeon HD 7750 in low profile form factor too, but the offer is a bit more limited.

Personally, I don't like low profile cards very much, since they tend to skimp on cooling (either small heatsinks with small fan, or medium-sized heatsinks and no fan) so you might even want to consider underclocking if the case ventilation is not exactly great (or if room temperature is much higher than 20C), or you risk burning it after heavy gaming sessions.
Of course, if you can find a low profile card with decent cooling, that's another story.

Due to VAT, generally a card priced at around 100 USD will be close in price to 100 EUR (for instance, here in Romania, it would be ~96 EUR) - not exactly the same price, but quite close, and after figuring in the usual retailer profit, you're more or less at the same price level.
NVIDIA has some decent contenders here and there around the 130, 170 and 300+ USD price levels as far as average raw FPS per price goes, but generally AMD is from ever so slightly better on average to noticeably better, due to the still-perceived "underdog" effect (less aggressive advertising).
Personally, I prefer NVIDIA because there's something annoying about AMD drivers I can't quite put a finger on, but whenever I have to buy budget, that's no longer a concern and I almost always go AMD.

Various games run better on AMD, others on NVIDIA, others pretty much the same, so depending on what you plan to play (if your selection is usually limited to games that perform much better on one), either one might end up being a better choice.
Factory-overclocked models vary greatly from manufacturer from manufacturer, and some can be pushed quite a bit above stock speed specifications (at a sacrifice in power consumption and heat levels).
NVIDIA also gives you PhysX (not that you will really notice though), but AMD gives you the ability to use 3 monitors at the same time on the cheaper cards too (NVIDIA only started doing that since the latest series, the 600s).

Also, your country matters a lot in terms of what will be available.
You will generally get a pretty wide selection in UK and Germany, but the selection will be far more limited in Italy, Bulgaria or Romania.

As for video RAM, 1GB should be more than enough for FullHD on two monitors at the same time for almost any game (at a good to best graphic detail level). Exceptions might of course exist, but it's usually fancy non-online games.
You really do not want the video card to start using system RAM for textures and such (not even sure all cards can do that), as it's a lot slower, and performance will degrade greatly.
Plus, you also want a decent amount of system RAM either way, you REALLY don't want to start getting pagefile usage while you're gaming, that's an additional horrible performance drop even if you are using a SSD, let alone a slower HDD.

Depending on which i5 exactly, the "utter trash" integrated GPU could be anywhere up to two times faster than a stock 8400 GS.
Again, a 8400 GS is the cheapest, lowest rung retail NVIDIA video card from 5 years ago.

You don't want to run it a minimum, that's for sure, it's horrible. Not unless you are doing WvW and it's the win that counts. I drop it to min in WvW at times, but outside of that you want at least mid settings imo.

Passmark ratings (averages ; higher is better):
* standalone GeForce 8400GS -> 162
* Intel HD (3000) from a i5-2430M -> 330

Well, not much of a surprise though, when you're looking at a gap of 5 generations and you're comparing the weakest of the standalone lineup from that time.
Look at GeForce 8800 GTX, which actually came out 7 months earlier, it has a passmark rating of 1045, higher than a Radeon HD 6570 which only gets 835... but the 6570 has a TDP of 44W, supports DX11, whereas the 8800GTX has a 155W TDP and only DX10.0 (not to speak of prices back in the day, the GTX would have sold even for over 500$ near launch, the AMD for just 55$.

You should see the passmark ratings on the last two AMD CPU/GPU combos too...

Desktop:
Radeon HD 6550D, inside A8-3800 through A8-3870K processors -> PM = 630
Radeon HD 7660D, inside the desktop A10-5700 and A10-5800K processors -> PM = 726
Mobile:
Radeon HD 6620G, inside A8-35*0MX line -> PM = 470
Radeon HD 7660G, inside the mobile A10-4600M -> PM = 617

And, if you are to discount some of the results that look too odd (like the hard to believe score of 1180 from a "Intel HD i7-2760QM"), the average over those reporting Intel HD 4000 (most likely desktop IvyBridge iGPUs) somewhere around 563.