From what I understand, the "Energy Star" rating tells you they will come on at full brightness (among other features). The only fluorescent bulbs I have found that take a minute or so to get to full brightness are the reflector-style bulbs (and they don't have the Energy Star rating). I think all the tube-type bulbs will come on fully bright.

I would suggest a couple of things for you. One is to consider whether you would be better sticking to conventional incandescent or halogen lighting, perhaps incandescent strip lighting. Full brightness instantly, and more importantly the colour of your clothes will appear correct. Fluorescents do not display colours completely accurately. If you have more clothes sense than me, you may be upset when that perfect match turns into a nasty clash as soon as you step outside. The downsides of the incandescent lighting are that it uses more energy and bulbs don't last as long. Neither should be an issue in a closet where they will be used for just a minute or two each day.

If you still want to try fluorescent tubes, they are all fairly bright as soon as you turn them on, nowhere near as bad as some CFLs. There may be a second or so as they flicker and get organised but you won't notice them getting much brighter after that. I still recommend you think carefully about the colour rendering issue, and look for tubes with a high CRI above 90. They are more expensive but worth it for this sort of application.

I don't believe that Energy Star mandates any particular requirements for initial brightness. There is a specification of less than one second from switching on to continuous operation of the light (ie. no flickering after 1 second). You can read the full specifications at the link.

Here is a link that might be useful: Energy Star residential requirements

As electricians, we have installed many flourescent fixtures in closets. We usually mount a one or two tube fixture on the inside wall between the top of the door and the ceiling with a switch outside of the closet. The length of the fixture depends on the size of the closet. This gives excellent lighting to the shelf and the clothes. Regarding electronic ballasts, they are as fast as an IC bulb, there is no flickering and there is no warm up time. Also, if these fixtures hum, it usually means they are not grounded properly.