Its not REALLY possible to increase the res of a digital image once its at a specific resolution. You can easily lower it but increasing is another matter. That being said there are a number of ways to simulate a higher res. The best and easiest, is to use a program that does this for you. In effect the program create new pixels based on existing ones and thus increase the apparent res. I use a program called PhotoZoom to do this. Its not the best on the market but its a good trade off between the best, which is rather expensive, and doing it yourself manually.
Manually: in some cases you can simulate a higher res as follows. Change the res under Image>Image Size to whatever you prefer, I would suspect the current res is 72 so increase it to 300 or even 600. If your image has multiple colours/elements (it does) create a duplicate layer for each colour/element. Start with a layer, say the one that will capture the green portions of your image and use the Gaussian Blur filter to blur the the image a bit. Using the magic wand tool (not constrained) select the green. Tweak the sensitivity settings of the wand so that your selection extends a bit into the fuzzy area around the solid colour. Using the eyedropper tool select for the green and then fill the selected area, now, invert your selection and clear everything but the green area. Repeat this for each solid colour in the image and then flatten the layers. You will now have what is in a respect a higher res image, that is, more dpis than you started with. The downside to this method is that it will not work with images that have many shades of colour (or grey) and it will reduce the sharpness of corners a bit. If this is a concern before blurring the image increase its size 500%, now do the blur etc and then reduce the size to about 35 or 40%. Sometimes I find I can reduce the size back to the original by inputing 20% but on occasion this produces unwanted artifacts.
Also, you will have a bit of an issue with the gold and green triangle portion of the image since these are not a solid colour.
In this case create 2 layers for the gold section and 2 for the green triangle, turn off the upper layer, treat the lower layer(s) as follows: clear everything but the green triangle. Select the empty background and run a narrow stroke around it, blur as above and select the edge, expand it by 1 2 or 3 pixels as nec, and do a content aware fill for it, then turn the upper layer back on and adjust the upper layers opacity to get as acceptable a result as possible. Since the selection has extended a bit beyond the edge of the original object you should have an acceptable blend with the solid upper layer providing the majority of the colour gradient and the lower layer providing for a fine edge (I hope).
You may have to play with this process a bit to get the best results possible especially since there are 2 portions that have internal shading.
If you wish I will be happy to run the image through PhotoZoom for you (you might be able to download a trial and do it yourself,,,,,I do not recall that PZ produced watermarks when I tested it but that was quite some time ago so it may do so now).
I hope this helps.
