Yes, you can load Cloudmade tiles into ArcGIS 10 by using ArcBrutile.
This open source extension comes with some map tile services of which Cloudmade is one.
You can see the xml config file of CloudMade with 3 styles at this [link]
<TileMap
title="Midnight Commander"
srs="EPSG:4326"
type="InvertedTMS"
profile="local"
href="http://dl.dropbox.com/u/9984329/ArcBruTile/Services/Cloudmade/Midnight Commander.xml" />
<TileMap
title="Pale Dawn"
srs="EPSG:4326"
type="InvertedTMS"
profile="local"
href="http://dl.dropbox.com/u/9984329/ArcBruTile/Services/Cloudmade/Pale Dawn.xml" />
<TileMap
title="Fresh"
srs="EPSG:4326"
type="InvertedTMS"
profile="local"
href="http://dl.dropbox.com/u/9984329/ArcBruTile/Services/Cloudmade/Fresh.xml" />
Opening one of those style links, you can see more details about configuration of the style, e.g Pale Dawn
<Title>OSM_New_cache</Title>
<Abstract></Abstract>
<SRS>EPSG:4326</SRS>
<BoundingBox minx="-180.000000" miny="-90.000000" maxx="180.000000" maxy="90.000000" />
<Origin x="-180.000000" y="-90.000000" />
<TileFormat width="256" height="256" mime-type="image/png" extension="png" />
<TileSets>
<TileSet href="http://c.tile.cloudmade.com/BC9A493B41014CAABB98F0471D759707/998/256/0"
units-per-pixel="0.70312500000000000000" order="0" />
<TileSet href="http://c.tile.cloudmade.com/BC9A493B41014CAABB98F0471D759707/998/256/1"
units-per-pixel="0.35156250000000000000" order="1" />
<TileSet href="http://c.tile.cloudmade.com/BC9A493B41014CAABB98F0471D759707/998/256/2"
units-per-pixel="0.17578125000000000000" order="2" />
<TileSet href="http://c.tile.cloudmade.com/BC9A493B41014CAABB98F0471D759707/998/256/3"
units-per-pixel="0.08789062500000000000" order="3" />
<TileSet href="http://c.tile.cloudmade.com/BC9A493B41014CAABB98F0471D759707/998/256/4"
units-per-pixel="0.04394531250000000000" order="4" />
<TileSet href="http://c.tile.cloudmade.com/BC9A493B41014CAABB98F0471D759707/998/256/5"
units-per-pixel="0.02197265625000000000" order="5" />
<TileSet href="http://c.tile.cloudmade.com/BC9A493B41014CAABB98F0471D759707/998/256/6"
units-per-pixel="0.01098632812500000000" order="6" />
<TileSet href="http://c.tile.cloudmade.com/BC9A493B41014CAABB98F0471D759707/998/256/7"
units-per-pixel="0.00549316406250000000" order="7" />
<TileSet href="http://c.tile.cloudmade.com/BC9A493B41014CAABB98F0471D759707/998/256/8"
units-per-pixel="0.00274658203125000000" order="8" />
<TileSet href="http://c.tile.cloudmade.com/BC9A493B41014CAABB98F0471D759707/998/256/9"
units-per-pixel="0.00137329101562500000" order="9" />
<TileSet href="http://c.tile.cloudmade.com/BC9A493B41014CAABB98F0471D759707/998/256/10"
units-per-pixel="0.00068664550781250000" order="10" />
<TileSet href="http://c.tile.cloudmade.com/BC9A493B41014CAABB98F0471D759707/998/256/11"
units-per-pixel="0.00034332275390625000" order="11" />
<TileSet href="http://c.tile.cloudmade.com/BC9A493B41014CAABB98F0471D759707/998/256/12"
units-per-pixel="0.00017166137695312500" order="12" />
<TileSet href="http://c.tile.cloudmade.com/BC9A493B41014CAABB98F0471D759707/998/256/13"
units-per-pixel="0.00008583068847656250" order="13" />
<TileSet href="http://c.tile.cloudmade.com/BC9A493B41014CAABB98F0471D759707/998/256/14"
units-per-pixel="0.00004291534423828125" order="14" />
<TileSet href="http://c.tile.cloudmade.com/BC9A493B41014CAABB98F0471D759707/998/256/15"
units-per-pixel="0.00002145767211914062" order="15" />
<TileSet href="http://c.tile.cloudmade.com/BC9A493B41014CAABB98F0471D759707/998/256/16"
units-per-pixel="0.00001072883605957031" order="16" />
<TileSet href="http://c.tile.cloudmade.com/BC9A493B41014CAABB98F0471D759707/998/256/17"
units-per-pixel="0.00000536441802978516" order="17" />
<TileSet href="http://c.tile.cloudmade.com/BC9A493B41014CAABB98F0471D759707/998/256/18"
units-per-pixel="0.00000268220901489258" order="18" />
<TileSet href="http://c.tile.cloudmade.com/BC9A493B41014CAABB98F0471D759707/998/256/19"
units-per-pixel="0.00000134110450744629" order="19" />
</TileSets>
</TileMap>
And then you should find document of CloudMade tiles at here to understand what BC9A493B41014CAABB98F0471D759707/998/256 is.
Finally, to have your custom style loaded into ArcGIS, you can use Firebug on Firefox to inspect structure of the custom tiles. (! i have not used CloudMade for a long time so i dont remember whether CloudMade show you directly schema of custom tiles or not, but Firebug should work as i had a test).
The final step, writing another Cloudemade xml is not a big deal for you..i think.
Hope this helps.