Looking for a graphics card for the Dell Optiplex 780 MT

4GB? I think not, but there are 2GB R7 260X cards for around $90 that will be the best for that price range. the R7 260X take a six pin connector for power, you will most likely need a new power supply, also due to the motherboard layout in optiplexes of that time, you will probably need a PCIE riser.

This is Genuine Dell Dual PCI Riser Card Assembly
I am not sure that OP can use RISER with Mini Tower, back panel has no removable horizontal parts, unlike Desktop, unless OP would have to use cable RISER and use scotch tape to secure card in the case - too much DIY work, Dremel might be necessary.
Also, Dell RISER accepts only single bracket GPU




This is html manual for Optiplex 780 DT http://ebay.redplanettrading.com/desktops/case/optiplex780/Optiplex780_SM_DT/riser_cage.htm

This is back panel of Optiplex 755 DT with RISER (very similar in construction to Optiplex 780 DT)



This is back panel of Optiplex 780 DT, please excuse me for a large picture



So Dell RISER can be used only for Desktop Optiplexes not MINI TOWER.

Just got confirmation that AMD UEFI based cards are not compatible with LEGACY based Dells http://www.tomshardware.com/forum/id-2755310/dual-uefi-legacy-bios-video-cards.html#16645057 do to the end of the thread.
I wonder which BIOS Optiplex 780 has?

Lucky one, it is UEFI



But still RISER question.

Be aware that the Dell Optiplex mainboards are built the wrong way round. So you can't use the neighbouring slot for a dual hight card.
The Optiplex 780 MT does not have low profil. Only the DT and the SFF versions have.

In any case you have to take the CPU heatsink into consideration, because it blocks the way for longer cards, if the GPU heatsink is higher than one slot. Another isue is, that the pcie 16x slot is only connected with one lane and not with 16. So you probably won't get any high frame rates if you plan on using it for games, but i'm no expert there.

I have an Optiplex 780 DT. After a lot of searching i finally put in a Zotac GTX 750 TI LP with a riser card. It also works as LP, the brackets are included. And you don't need additional power (It uses less than 75W). Unfortunately it is no longer available. I only find the GT 740 LP. All better Cards were longer and had too high heatsinks. But as i don't use it for games, it gives me the third screen and it plays 1080 videos without any missing frames. In some cases it even plays 2160, but that depends on the framerate, the bitrate and the codec of the Video.

If you have a MT version you get other cards in, as long as they are only 1 Slot high (including the heatsink) or they are 1 slot high (with a higher heatsink and at least 1mm shorter than the original ATI Radeon card. Since i also have the MT version here I just checked.
For example the other Zotac GTX 750 TI versions (which seem to be still available at the moment) should work in the MT case (excluding the ZOTAC GTX 750 Ti OC which is too long), if you cut the upper part of the bracket away. But I think the won't be available much longer. I think I'm just going to order one for the MT case.

You might find similar cards with the same dimensions.

I hope that helps

Update:
I did another search. For the MT case, the MSI R7 360 2GD5 OC should work too, if you cut the upper part of the bracket. You might have to cut of a bit of the heatsink cover too, but I'm not shure. It might fit there. Because I never had such a card, i can't tell you if it works.
The only cards I found, that meet your technical specs and should fit in the case, are the Radeon R9 Nano ones. But they cost about 5 or 6 times as much. So if the 4GB are necessary, i would buy another computer with a mainboard and case that meets the ATX standard. There you can fit in any kind of card. If you keep the harddrive, optical drive and RAM it's cheaper too.

BTW: My Dell Optiplex 780 runs with 16GB RAM.