Performance-wise, you would be fine with anything like the GT 1030, RX 460/550/560, or GTX 1050/1050TI (http://www.tomshardware.com/reviews/nvidia-geforce-gt-1030-2gb,5110-6.html).
Power-wise, I'm thinking that you're looking at being limited to the GT 1030 or GTX 1050. The 1050TI models usually require PCIe power connectors (which would require a new PSU), & I think the same applies to the RX 460/560 cards...& I'm not sure the RX 550 is low-powered enough either.
I'm assuming you mean that your budget is from $80 to $140 USD. Based on that, both the GT 1030 & GTX 1050 should fit into your price range (GT 1030 on the low, GTX 1050 on the high):
https://pcpartpicker.com/products/video-card/#c=396,379,380&sort=price
To be honest, since it's DotA 2, & if your monitor is truly limited to the 800x600 resolution you have listed there. I wouldn't bother with the 1050. Just save some money & get the GT 1030 for now.