In what concerns Busemann or horofunctions (I prefer to call them Busemann cocycles, because in invariant language they are cocycles, not functions) for symmetric spaces, there are several sources of various degree of explicitness.

(1) For the group $SL(d,\mathbb R)$ (and the associated symmetric space) theBusemann cocycle essentially appears in Furstenberg's formula for the rate ofgrowth of random matrix products, see

MR0163345 (29 #648) Furstenberg, HarryNoncommuting random products. Trans. Amer. Math. Soc. 108 1963 377--428.

From the geometrical point of view the main idea there is that if you want to find the linear rate of growth of $d(o,g_1 g_2\dots g_n o)$, where $o$ is a reference point, and $g_i$ is a stationary sequence of random isometries, then you can look at the increment $d(o,g_1 g_2\dots g_n o)-d(o,g_2 g_3\dots g_n o)=d(g_1^{-1}o,g_2g_3\dots g_n o)-d(o,g_2 g_3\dots g_n o)$, which converges to the Busemann cocycle $\beta_\gamma(o,g^{-1}o)$ provided $g_2g_3\dots g_n o$ converges to a boundary point $\gamma$ in the Busemann compactification. The cocycle itself looks, roughly speaking, like $\log \|gv\|/\|v\|$, where $g$ is the matrix (or its exterior power) representing a point in the symmetric space, and $v$ is a vector representing the boundary point. There is also a lot about it in later papers by Guivarc'h.

(2) Busemann cocycles naturally appear in various works related to compactifications of symmetric spaces. Historically the first source is the monograph of Karpelevich

MR0231321 (37 #6876) Karpelevic, F. I. The geometry of geodesics and the eigenfunctions of the Beltrami-Laplace operator on symmetric spaces. Trudy Moskov. Mat. Obšc. 14 48--185 (Russian); translated as Trans. Moscow Math.Soc. 1965 1967 pp. 51--199. Amer. Math. Soc., Providence, R.I., 1967.

where he explicitly discusses pencils of convergent geodesics and introduces the associated horospheric coordinates. Later expositions are in two books on compactifications of symmetric spaces:

MR1633171 (2000c:31006) Guivarc'h, Yves(F-RENNB-IM); Ji, Lizhen(1-MI); Taylor, J. C.(3-MGL)Compactifications of symmetric spaces. (English summary) Progress in Mathematics, 156. Birkhäuser Boston, Inc., Boston, MA, 1998. xiv+284 pp. ISBN: 0-8176-3899-7

and

MR2189882 (2007d:22030) Borel, Armand(1-IASP); Ji, Lizhen(1-MI)Compactifications of symmetric and locally symmetric spaces. Mathematics: Theory \& Applications. Birkhäuser Boston, Inc., Boston, MA, 2006. xvi+479 pp. ISBN: 978-0-8176-3247-2; 0-8176-3247-6

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Of course, the first example should be non-compact Riemannian symmetric spaces, where the Busemann (horofunctions in your terminology) functions are known in a pretty explicit form. I don't think there are other explicit examples.